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Philosophical transactions of the 
Royal Society of London 

Royal Society (Great Britain), JSTOR (Orgariizatiori) 








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R 



PHILOSOPHICAL 

TRANSACTIONS. 

/ {^ ■' ' :i 

OF THE 

ROYAL SOCIETY OF LONDON. 



Seiiies a. 

CONTAINING PAPEES OF A MATHEMATICAL OK PHYSICAL CHAKACTEE. 



VOL. 195. 



LONDON: 

IMJfNTED I5Y IIARKISON AM) SONS, ST. MARTINS LANB, W.O., 
|)rinttTS in #rbinBni lo $ti Pajtsig. 

January, 1901, 



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C m ] 



CONTENTS. 

(A) 
VOL. 195. 



List of Illustrations page ▼ 

Advertisement vii 

List of Institutions entitled to receive the Philosophical Transactions or Proceedings of the Royal 

Society ' ix 

Adjudication of Medals xvii 



X T. Mathematical Coritribntions to the Theory of Evolution, — VIl. On the Con'elatio7i 
of Characters not Quantitatively Measurable. By Karl Peaiison, F.Ii.S. 

page 1 

11. Electrical Conductivity in Gases Traversed by Cathode Rays. By J. C. 
McLennan, Demonstrator in Physics^ University of Toronto. Communicated 
by Professor J. J. Thomson, F.R.S. 49 

^ 111. Mathematical Contnbutions to the Theory of Evolution. — VIII. On the Inheri- 
tance of Character's not capable of Exact Qvxintitative Meamrement. — 
Part I. Introductory. Part II. On the Inheritance of Coat-colour in Horses. 
Part III. On the Inheritance of Eye-colour in Man. By Kabl Peabson, 
F.R.S. , ivith the assistance of Alice Lee, D.Sc.y University College, 
London 79 

IV. On Simultaneous Partial Differential Equations. By A, C. Dixon, Sc.D. 

Communicoied by J. W. L. Glaibheu, Sc.D 151 

a 2 



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[ iv ] 

V. The Velocity of the Ions produced in Gases hy Ronlgen Rays. By John Zeleny, 

B.Sc.y B.A., Assistant Professor of Physics, University of Minnesota, Com- 
municated hy Professor J. J. Thomson, F.R.S. 193 

VI. Underground Temperature at Oxford in the Yexxr 1899, as determined hy Five 

Platinum-resistance Thermometers. By Arthur A. Rambaut, M.A., D.Sc, 
Radcliffe Ohserver. Communicated hy E. H. Griffiths, F.R.S. . . . 235 

VII. The Diffusion of Ions produced in Air hy the Action of a Radio-active 

SuhstancCy Ultra-violet Light and Point Discharges. By John S. Townsend, 
M.A., Clerk-Maxwell Student, Cavendish Laboratory^ Fellow of Trinity 
College^ Cambridge. Communicated hy Professor J. J. Thomson, F.R,S. 259 

VIII. T/ie Crystalline Structure of Metals. (Second Paper.) By J. A. EwiNG, 
F.R.S. , Professor of Mechanism and Applied Mechanics in the University of 
Cambridge, and Walter Rosenhain, B.A., St. John's College, Cambridge^ 
\^5\ Exhibition Research Scholar, University of Melbourne .... 279 

IX. Lilies of Induction in a Magnetic Field. By Professor H. S. Hele-Shaw, 

LL.D., F.R.S., and Alfred Hay, B.Sc 303 

X. On tlie Application of Fourier's Double Integrals to Optical Problems. By 

Charles GtODFREY, B.A., Scholar of Trinity, Isaac Neioton Student in the 
University of Cambridge. Communicated by Professor J. J. Thomson, 
F.R.S. 329 

XL An Experimental Investigation into the Flow of Marble. By Frank Dawson 
Adams, M.Sc, Ph.D., F.G.S., Logan Professor oj Geology in McGill 
University, and John Thomas Nicolson, D.Sc, M.Inst., C.E., Head of the 
Engineering Departmenty Manchester Municipal Technical School {formerly 
Professor of Mechanical Engi7ieeri7ig in McGill University) .... 363 

Index to Volume 403 

Erratum 405 



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LIST OF ILLUSTRATIONS. 

Plates* 1 and 2. — Dr. A. A. Rambaut on the Underground Temperature at Oxford 
in the year 1899, as determined by Five Platinum-resistance Thermometers. 

Plates 3 to 13. — Professor J. A. Ewing and Mr. W. Rosenhain on the Crystalline 
Structure of Metals. 

Plates 14 to 21. — Professor H. S. Hele-Shaw and Mr. Alfred Hay on Lines of 
Induction in a Magnetic Field. 

Plates 22 to 25. — Professor F. D. Adams and Dr. J. T. Nioolson on an Experimental 
Investigation into the Flow of Marble. 



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[ vii ] 



ADVERTISEMENT, 



The Committee appointed by the Royal Society to direct the publication of the 
Philosophical Transactions take this opportunity to acquaint the public that it fuDy 
appears^ as well from the Council-books and Journals of the Society as from repeated 
declarations which have been made in several former Transactions^ that the printing of 
them was always, from time to time, the single act of the respective Secretaries till 
the Forty-seventh Volume ; the Society, as a Body, never interesting themselves any 
further in their publication than by occasionally recommending the revival of them to 
some of their Secretaries, when, from the particular circumstances of their affairs, the 
Transactions had happened for any length of time to be intermitted. And this seems 
principally to have been done with a view to satisfy the public that their usual 
meetings were then continued, for the improvement of knowledge and benefit of 
mankind : the great ends of their first institution by the Royal Charters, and which 
they have ever since steadily pursued. 

But the Society being of late years greatly enlarged, and their communications more 
numerous, it was thought advisable that a Committee ot their members should be 
appointed to reconsider the papers read before them, and select out of them such as 
they should judge most proper for publication in the future Transactions ; which was 
accordingly done upon the 26th of March, 1752. And the grounds of their choice are, 
and will continue to be, the importance and singularity of the subjects, or the 
advantageous manner of treating them ; without pretending to answer for the 
certainty of the facts, or propriety of the reasonings contained in the several papers 
so published, which must still rest on the credit or judgment of their respective 
authors. 

It is likewise necessary on this occasion to remark, that it is an established rule of 
the Society, to which they will always adhere, never to give their opinion, as a Body, 



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[ ^^iii ] 

upon any subject, either of Nature or Art, that comes before them. And therefore the 
thanks, which are frequently proposed from the Chair, to be given to the authors of 
such papers as are read at their accustomed meetings, or to the persons through whose 
hands they received them, are to be considered in no other light than as a matter of 
civility, in return for the respect shown to the Society by those communications. The 
like also is to be said with regard to the several projects, inventions, and curiosities of 
various kinds, which are often exhibited to the Society ; the authors whereof, or those 
who exhibit them, frequently take the liberty to report, and even to certify in the 
public newspapers, that they have met with the highest applause and approbation. 
And therefore it is hoped that no regard will hereafter be paid to such reports and 
public notices ; which in some instances have been too lightly credited, to the 
dishonour of the Society. 



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[ ix ] 
1900. 



List of Institutions entitled to receive the Philosophical Transacteons or 

Proceedings of the Royal Society. 



Institutions marked a are entitled to receive Philosophical Transactions, Series A, and Proceedings. 
„ „ B „ „ „ „ Series B, and Proceedings. 

„ AB „ „ „ „ Series A and li, and Procee<ling«. 

„ „ /> „ „ Proceedings only. 



America (Central). 
Mexico. 

p. Sociedad Cientifica " Antonio Alzate." 
Ainerica(North). (See United States and Canada.) 
America (South). 
Buenos Ayres. 

A6. Mnseo Naclonal. 
Caracas. 

B. University Library 
Cordova. 

AB. Academia Nacional de Ciencias. 
Demerara. 
p, Hoyal Agricultural and Commercial 
Society, British Guiana. 
LiQ. Plata. 

B. Museo de La Plata. 
Rio de Janeiro. 
p. Observatorio. 
Australia. 
Adelaide. 

p. Royal Society of South Australia. 

Brisbane. 

^. Royal Society of Queensland. 
Melbourne. 

p. Observatory. 

p. Royal Society of Victoria. 

AB. University Library. 
Sydney. 



Australian Museum. 

Geological Survey. 

Linnean Society of New South Wales. 

Royal Society of New South Wales. 

University Library. 



P- 
P- 
P- 

AB. 
AB* 

Austria. 

Agram. 

p, Jugoslavenska Akademija Znanosti i Um< 
jetnosti. 

p. Societas Historico-Naturalis Croatica. I 

VOL. CXCV. — A. b 



Anstria (continued). 
Briinn. 

AB. Naturforachender Verein. 
Gratz. 
AB. Naturwissenschaftlicher Verein fur Steier- 
mark. 
Innsbinick. 

AB. Das Ferdinandeum. 

p, Naturwissenschaftlich - Medicinischer 
Verein. 
Prague. 

AB. Konigliche Bohmisclie Gesellschaft dcr 
Wissenschaften. 
Trieste. 

B. Museo di Storia Naturale. 
p. Societa Adriatica di Scienze Naturali. 
Vienna. 



Antbropologische Gesellschaft. 
Kaiserliche Akademie der Wissenschaften. 
K.K. Geographische Gesellschaft. 
K.K. Geologische Reichsanstalt. 
K.K. Naturhistorisches Hof- Museum. 
K.K. Zoologisch-Botanische Gesellschaft. 
Oesterreichische Gesellschaft fiir Meteoro- 

logie. 
Von Kuffner'sche Stemwarte. 
Belgium. 
Brussels. 

B. Academic Roy ale de Medecine. 
Academic Royale des Sciences. 
Mus^ Royal d'Histoire Naturelle de 

Belgique. 
Observatoire Royal. 
Soci6t6 Beige de Geologic, de Paleonto- 

logie, et d'Hydrologie. 
Societe Malacologique de Belgique. 



P- 

aB. 

P' 

AB. 
B. 
B. 
P' 

A. 



AB. 



P- 
P' 



P' 

Ghent. 
AB. Univei-site. 



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\.. 



[ X J 



Belgium (continued). 
Li^ge. 

AB. Soci^te des Sciences. 

p. Soci^te Geologique de Belgique, 
Lonvain. 

B. Laboratoire de Microscopic et de Biologie 
Cellulaire, 

AB. Universite. 
Canada. 

Predericton, N.B. 

p, Universitj of New Brunswick. 
Halifax, N.S. 

p. Nova Scotian Institute of Science. 
Hamilton. 

p, Hamilton Association. 
Montreal. 

AB. McGill University. 

p. Natural History Society. 
Ottawa. 

AB. Geological Survey of Canada. 

AB. Royal Society of Canada. 
St. John, N.B. 

p. Natural History Society. 
Toronto. 

p. Toronto Astronomical Society. 

p, Canadian Institute. 

AB. University. 
Windsor, N.S. 

p. King's College Libi*ary. 
Cape of Oood Hope, 

A. Observatory. 

AB. South African Library. 
Ceylon. 
Colombo. 

B. Museum. 
China. 

Shanghai. 
p. China Branch of tlie Royal Asiatic Society. 

Denmark. 
Copenhagen. 

AB. Kongelige Danske Videnskabernes Selskab. 
Egypt. 

Alexandria. 

AB. Biblioth^que Municipale. 
England and Wales. 
Aberysiwith. 

AB. Univei-sity College. 
Bangor. 

AB. University College of Noith Wales. 
Birmingham. 

AB. Free Central Library. 

AB. Mason College. 

p. Philcsophical Society. 



England and Wales (continued). 
Bolton. 

p. Public Library. 
Bristol. 

p. Merchant Venturers' School. 

AB. Univei*8ity College. 
Cambridge. 

AB. Philosophical Society. 

p. Union Society. 
Cardiff. 

AB. University College. 
Cooper's Hill. 

AB. Royal Indian P]ngineering College. 
Dudley. 

p, Dudley and Midland Geological aud 
Scientific Society. 
Essex. 

p. Essex Field Club. 
Falmouth. 

p. Royal Cornwall Polytechnic Society. 
Greenwich. 

A. Royal Observatory. 
Kew. 

B. Royal Gardens. 
Leeds. 

p. Philosophical Society, 

AB. Yorkshire College. 
Liverpool. 

AB. Free Public Library. 

p, Literary and Philosophical Society. 

A. Observatory. 

AB. University College. 
London. 

AB. Admiralty. 

p. Anthropological Institute. 

AB. Board of Trade (Electrical Standards 
Laboratory). 

AB. British Museum (Nat. Hist.). 

AB. Chemical Society. 

A. City and Guilds of London Institute. 
p. '* Electrician," Editor of the. 

B. Entomological Society. 
AB. Geological Society. 

AB. Geological Survey of Gi'eat Britain. 

p. Geologists' Association. 

AB. Guildhall Library. 

A. Institution of Civil Engineers. 

p. Institution of Electrical Engineers. 

A. Institution of Mechanical Engineers. 

A. Institution of Naval Architects. 
p. Iron and Steel Institute. 

AB. King's College. 

B. Linnean Society. 
AB. London Institution. 



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[ xi ] 



, 



f 

1 



^' 



England and Wales (continued). 
London (continued). 

London Library. 

Mathematical Society. 

Meteorological Office. 

Odontological Society. 

Pharmaceutical Society. 

Physical Society. 

Quckett Microscopical Club. 

Royal Agricultural Society. 

Royal Astronomical Society. 

Royal College of Physicians. 

Royal College of Surgeons. 

Royal Eiigiueera (for Libraries abroad, six 

copies). 
Royal Engineers. Head Quarters Library. 
Royal Geographical Society. 
Royal Horticultural Society. 
Royal Institute of British Architects. 
Royal Listitution of Great Britain. 
Royal Medical and Chirurgical Society. 
Royal Meteorological Society. 
Royal Microscopical Society. 
Royal Statistical Society. 
Royal United Service Institution. 
Society of Arts. 

Society of Biblical Archaeology. 
Society of Chemical Industry (London 

Section). 
Standard Weights and Measures Depaii- 

ment. 
The Queen's Library. 
The War Office. 
University College. 
Victoria Institute. 
Zoological Society. 
Manchester. 

AB. Free Library. 

AB. Literary and Philosophical Society. 

p. Geological Society. 

AB. Owens College. 

Netley. 

p. Royal Victoria Hospital. 

Newcastle. 

Free Library. 

North of England Institute of Mining and 
Mechanical Engineers. 
p. Society of Chemical Industry (Newcastle 
Section) . 
Norwich. 

p, Norfolk and Norwich Literary Institution. 
Nottingham. 

AB. Fi'ee Public Library. 



A. 

P' 
P' 

V' 

V- 

p- 
p- 

A. 
B. 
B. 



AB. 

P' 

P- 

P^ 
AB. 

B. 
P' 
P' 
P- 

AB. 

AB. 

P- 

P' 



AB. 
AB. 
AB. 

B. 



AB. 

P' 



England and Wales ^continued). 
Oxford. 
p, Ashmolean Society. 
AB. Radcliffe Library. 

A. Radcliffe Observatory. 
Penzance. 

p. Geological Society of Cornwall. 
Plymouth. 

B. Marine Biological Aflsociation. 
p. Plymouth Institution. 

Richmond. 

A. " Kew " Observatory. 
Salford. 

p. Royal Museum and Library. 
Stonyhurst. 

p. The College. 
Swansea. 

AB. Royal Institution. 
Woolwich. 

AB. Royal Artillery Library. 
Finland. 
Helsingfors. 

p. Societas pro Fauna et Flora Fennica. 

AB. Soci^te des Sciences. 
France. 
Bordeaux. 



P- 
V' 
P- 
P- 



Academic des Sciences. 
Faculte des Sciences. 
Soci6t^ de M6decine et de Chirurgie. 
Soci^t^ des Sciences Physiques et 
Naturelles. 
Caen. 

p. Soci^t^ Linn6enne de Normandie. 
Cherbourg. 

p. Soci6te des Sciences Naturelles. 
Dijon. 

p. Academic des Sciences. 
Lille. 

p. Faculty des Sciences. 
Lyons. 
AB. Academic des Sciences, Belles - Let tres et Arts. 
AB. University. 
Marseilles. 

AB. Faculte des Sciences. 
Montpellier. 

AB. Academic des Sciences et Lettres. 
B. Faculty de M6decine. 
Nantes. 



P- 
Paris. 

AB. 
P- 

P' 



Soci^te des Sciences Naturelles de TOuest 
de la France. 

Academic des Sciences de Tlnstitut. 
Association Fran9ai8e pour TAvancement 

des Sciences. 
Bureau des Longitudes. 



h 2 



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[ xvi ] 



Qnited States (continued). 
Philadelphia. 

AB. Academy of Natural Sciences. 

AB. American Philosophical Society. 

p. Franklin Institute. 

p. Wagner Free Institute of Science. 
Rochester (N.Y.). 

p. Academy of Science. 
St. Louis. 

p. Academy* of Science. 
Salem (Mass.). 

p. American Association for the Adyance- 
ment of Science. 

AB. Essex Institute. 
San Francisco. 

AB. California Academy of Sciences. 



United States (continued). 
Washington. 
AB. Patent Office. 

Smithsonian Institution. 

United States Coast Surrey. 

United States Commission of Fish and 

Fisheries. 
United States Geological Survey. 
United States Nayal Observatory. 
United States Department of Agriculture. 
United States Department of Agriculture 
(Weather Bureau). 
West Point (N.Y.) 

AB. United States Military Academy. 



AB. 
AB. 
B. 

AB. 

AB. 

P' 
A. 



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I xvii ] 



Adjudication of the Medals of the Royal Society for the year 1900, 

by the President and Council. 



The COPLEY MEDAL to Professor Marcellin Berthelot, For. Mem.R.S., for 
his brilliant services to Chemical Science. 

The RUMFORD MEDAL to Professor Antoine Henri Becquerel, for his 
discoveries in Radiation proceeding from Uranium. 

A ROYAL MEDAL to Major Percy Alexander MacMahon, F.R.S., for the 
number and range of his contributions to Mathematical Science. 

A ROYAL MEDAL to Professor Alfred Newton, F.R.S., for his eminent 
contributions to the science of Ornithology and the Geographical Distribution of 
Animals. 

The DAVY MEDAL to Professor Guglielmo Koerner, for his brilliant investi- 
gations on the Position Theory of the Aromatic Compounds. 

The DARWIN MEDAL to Professor Ernst Haeckel, for his long-continued 
and highly-important work in Zoology, all of which has been inspired by the spirit 
of Darwinism. 



The Bakerian Lecture for the year 1900, " On the Specific Heat of Metals 
and the Relation of Specific Heat to Atomic Weight," was delivered by Professor W. 
A. Tilden, F.R.S., on March 8, 1900. 

The Croonian Lecture for the year 1900, " On Immunity with Special Reference 
to Cell Life," was delivered by Professor Dr. Paul Ehrlich, on March 22, 1900. 

VOL. CXCV. — A. C 



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PHILOSOPHICAL TRANSACTIONS. 



I. Mathematical Contributions to the Theoiy of Evolution. — VII. On the 
Correlation of Characters not Quantitatively Measurable. 

By Karl Pearson, F.R.S. 

{From tlie Deparimeitt of Applied Mathematics^ University College^ London,) 

Eeceived February 7,— Read March 1, 1900. 

NOTE. 

In August, 1899, I presented a memoir to the Boyal Society on the inheritance of coat-colour in the 
horse and of eye-colour in man, which was read November, 1899, and ultimately ordered to be published in 
the * Phil. Trans.' Before that memoir was printed, Mr. Yule's valuable memoir on Association was read, 
and, further, Mr. Lesue Brahley-Moore showed me that the theory of my memoir as given in § 6 of the 
present memoir led to somewhat divergent results according to the methods of proportioning adopted. 
We therefore undertook a new investigation of the theory of the whole subject, which is embodied in the 
present memoir. The data involved in the paper on coat-colour in horses and eye-colour in man have all 
been recalculated, and that paper is nearly ready for presentation.^ But it seemed best to separate the 
purely theoretical considerations from their application to special cases of inheritance, and accordingly the 
old memoir now reappears in two sections. The theory discussed in this paper was, further, the basis of a 
paper on the Law of Eeversion with special reference to the Inheritance of Coat-colour in Basset Hoimds 
recently communicated to the Society, and about to appear in the * Proceedings.*! 

While I am responsible for the general outlines of the present paper, the rough draft of it was 
taken up and carried on in leisure moments by Mr. Leslie Bramley-Moore, Mr. L. N. G. Filon, M.A., 
and Miss Alice Lee, D.Sc. Mr. Bramley-Moore discovered the w-functions ; Mr. Filon proved most of 
their general properties and the convergency of the series ; I alone am responsible for sections 4, 5, and 6. 
Mr. Leslie Bramley-Moore sent me, without proof, on the eve of his departure for the Cape, the 
general expansion for z on p. 26. I am responsible for the present proof and its applications. To Dr. 
Alice Lee we owe most of the illustrations and the table on p. 17. Thus the work is essentially a 
joint memoir in which we have equal part, and the use of the first personal pronoun is due to the fact 
that the material had to be put together and thrown into form by one of our number, — K P. 



Contents. 

page 
§ 1. On a General Theorem in Normal Correlation for two Variables. Series to Determine the 

Correlation 2 

§ 2. Other Series for the Determination of the Correlation 7 

* Since ordered to be printed in the * Phil. Trans.' 
t Bead January 25, 1900. ' Roy. Soc. Proc.,' vol. 66, p. 140. 
VOL. CXCV.— A 262. B 16.8.1900. 



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2 PROFESSOR K. PEARSON ON iMATHEMATICAL CONTRIBUTIONS 

§ 3. Proof of the General Convergency of the Series for the Correlation 10 

§ 4. On the Probable Error of the Correlation thus Determined 10 

§ 5. To Determine a Physical Meaning for the Series and on Divers Measures of Association ... 14 

§6. On the " Excess " and its Relation to Correlation and Relative Variability 18 

§ 7. On a Generalisation of the Fundamental Theorem of the present Memoir. Special Formulae 

for Triple and Quadruple Correlation 23 

§ 8. Illustrations of the Methods of the Memoir 35 

Illustration I. Inheritance of Coat-colour in Thoroughbred Horses. Sire and Filly 35 

„ II. Chance that an Exceptional Man is bom of an Exceptional Father 37 

„ III. Inheritance of Coat-colour in Dogs, Half-" Siblings " 38 

„ IV. Inheritance of Eye-colour between Maternal Grandmother and Granddaughter . . 39 

„ V. Inheritance of Statiu^e between Father and Son for different groupings 40 

„ VI. Correlation between Strength to resist Small-pox and Degree of Effective Vaccination 43 

„ VII. Effect of Antitoxin on Diphtheria Mortality 44 

„ VIII. Chance of Stock above the Average giving Produce above the Average as compared 

with the chance of such Produce from Stock below the Average 45 

„ IX. Chance of an Exceptional Man being bom of Exceptional Parents 46 



§ (1.) On a General Theorem in Normal Correlation. 
Let the frequency surface 



N 



z = 






where 



27rv/(l - r^)<ri<r^ 
N = total number of observations, 
cTi, 0*2 = standard deviations of organs x and y, 
r = correlation of x and y, 

be divided into four parts by two planes at right angles to the axes of x and y at 
distances h' and k' from the origin. The total volumes or frequencies in these parts 
will be represented by a, 6, c, and d in the manner indicated in the accompanymg 
plan : — 



TdiJb/e of Frequenc/ea 



dL 


b 


<SL'¥b 


C 


d 


c^d 


d*C 


b^d 


N 



Then clearly 



c? = 



27rv/(l 



^==- f fe- »i473(«' + 1" - ^^y) dxdy. . 

(1 - r»)J» Jt 



if 



27rv/(i - r») . 

h = h'/a-i and k = h'/tr^ 



(i.). 



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Further, 



and 



TO THE THEORY OF EVOLUTION, 



N_ r _ 



*^(fo; 



("•). 



^ V TT Jo 

<-^^ti)^^^=vif:e-% ■. . . .w. 

Thus, when a, 6, c, and dl are known, h and ifc can be found by the ordinary table of 
the probability integral, say that -of Mr. Sheppard (* Phil. Trans.,' A, vol. 192. p. 167, 
Table VI.*). The limits accordingly of the integral for d in (i.) are known. 

Now consider the expression 

.e-*rr^<*' + ^-^'^> = U,say, (vi.), 



x/1 - r« 
and let us expand it in powers of r. Then, if the expansion be 



/'d»U\ 



(vii.). 
(viii.). 



we shall have 

Taking logarithmic differentials, we get at once 

(1 _ r^)2^ = {xy + r(l - a^ - y^) + i^xy - 7^}JJ. 

Differentiating n times by Leibnitz's theorem, and putting r = 0, we have, after 
some reductions 

^«+i = ^(2n — 1 — x^ — y*)w^i 
- n{n —l){n — 2fun.^ 
+ ^{un + n{n — l)Un^^} (ix.). 



Hence we find 



Wo = 


= 1 






Mi = 


-xy 






«s = 


= («»- 


•1)(3^-1) 




^3 = 


= a;(a:»- 


-3)y(y«-3) 




W^r 


= («*- 


•6a:^ + 3)(y*-. 


6y« + 3) J 




* See, however, foot-note 


, p. 5. 






B 2 





(X.) 



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4 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 

Thus the following laws are indicated : — 

Un = v„X w„ (xL), 

where v, = an;,_i — (n — l)v<,_g (xiL), 

■w« = yw^-i — (« — l)«'«-9 (xiii.). 

We shall now show that these laws hold good by induction. . Assume 

Thus M,+i = xyu„ + n^u^_i — n(yw,v._i + aw,u>._j). 

But by (ix.), substituting for m,_3 from (xi.) and (xiii.), 

««+i = ay {v„w„ + w(n — 1)v,_2m;,_2J + n{2n —l—x^ — i/*)i;«_itp,_, 
— n{n — l)t\_iw,_i — ycyn{n — l)v„_8w;,_3 
+ n(n — 1) (yt',_iu»_2 + xVn^^Wn-i). 

4- n(n — 1) (y v„_iW,_s + a5t;,_gw,_i) 

+ a;u>,_i(a;v,_i — n— lv«_8} 

= v»+iW,+i, as we have seen above. 
Thus, if the theorem holds for u^, it holds for m,+i. Accordingly 

where the 7/s and v/b are given by (x.), (xiL), and (xiii.). 

It is thus clear that k~ \ I V dx dy consists of a series of which the general 



term is 

1 



n 



V«W«r* 



1 f* 
where V„ = -y--= e"**'v„da; 



'^'''■^\y-^«-<^- 



It remains to find these integrals. 
The general form of v^ is given by 



.. ==af-'^%^-^af-^+ "<- - '^%-, ^)<" - ^> af-* - &c. . . (xv.). 



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TO THE THEORY OF EVOLUTION. 5 

For this obviously gives (x.). Assume it true for v«_i and v„_2, then 

— XT 2U ^ + 22|2^ * — . . . 

= «». 
Thus the expression (xv.) is shown to hold by induction, the general terms being 

or the general term in u^. 



We notice at once that 

dx 



^r = ^^«-i (^^0- 



Thus, by (xii.) 

dx 



Vn — XVn^^ — 



Multiply by e"*"^ and integrate 

Integrating the latter integral by parts, we have 

Now y— e"*** can be found from any table of the ordinates of the normal curve, 

e.g., Mr. Sheppakd's, * Phil. Trans.,' A, vol. 192, p. 153, Table I.* We shall accord- 
ingly put 

= = C4«-'" K=^eH' (xvu.), 

and look upon H and K as known quantities. 

* For our present purposes the differences of Mr. Sueppa.rd's tables are occasionally too large, but the 
following series give very close results : — 

Let '^^ = Vl (<.-«• c)- (6 + d) ^ I V,^ ^ ^y (j^ )^ 



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6 PEOFESSOR K. PEAESON ON MATHEMATICAL C50NTRIBUTI0NS 

Further, let us write (v«-i)* » * as v«. j, and similarly (M;«_j)y » * as Ww.,. Thus 

V, = H.'U«.„ W, = K.'M>,_i (xviil). 

We have then from (i.) 

= ill \y''''''<^dy + s(|hk^,.,^._,) 

(ft + d)(<! + rf) "/r" - - \ 

by (ii.) and (iii.). 

Or, remembering that N = a + 6 + c4-c?>we can write this 

ad — he 2 /r"- - \ 

= r + ^ M + J (A« - 1) (i« - 1) + gA(A« - 3)*(i« -3) 

+ r^(A« - 15¥ + 45^2 - 15) (Jfc» - 15;fc* + 45*2 _ 15) 



5040' 
40320' 



+ ifti^A(A« - 21^* + 105A« - 105)ife(P - 21^ + 105*^- 105) + , &c. 

. . . . (xix,). 



Then A = xi + |lx.« + ^Xi' + ^Xi' + 



and 



1 ,75--/, 1 2.7 . , 127 . , 

H = ^^"i^ ■*• [2;^^ + [4 >^^ + 76->^^ + ■ 

1 7 127 

* = X2 + j3 X2» + jg- X2* + -ry-X2^ + • • • 

1 Ts-/, 1 Ys 7 . 127 . 



These follow from the considerations that if 



Xi = V2«-<^. 


Xj = V2t*s. 


d<h „ 


t=K. 


iH . 
#1 == - *' 





whence it is easy to find the successive differentials of h with regard to <f>i and k with regard to <^2> ^nd 

then obtain the above residts by Maclaurin's theorem. There is, of course, no difficulty in calculating % 

H and K from (xvii.) directly. That method was adopted in the niunerical illustrations. j 



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TO THE THEORY OF EVOLUTION. 7 

Here the left-hand side is known, and since h and k are known, we can find the 
coefficients of any number of powers of r so soon as the first two have been found, 
firom (xii.) and (xiii.). 

Accordingly the correlation can be found if we have only made a grouping of our 
frequencies into the four divisions, a, 6, c, and d. 

If h and k be zero, we have fi:om (xvii.) and (iv.) 

H = K= ^ 



i/27r 
The right-hand side of (xix.) is now ^ . c - - it? ^ ^ 



r+f3r» + 



5o <x ■ (L a**^ t - K,. 



or equal to sin"^ r. 



_. . ^ (ad — 6c) 

Hence r — sm 2ir ^^ — rr^ — 

= cos IT —2 (XX.), 

which agrees with a result of Mr. Sheppard's, * Phil. Trans.,' A, voL 192, p. 141. We 
have accordingly reached a generalised form of his result for any class-index whatever. 
Clearly, also, r being known, we can at once calculate the fi:equency of pairs of organs 
with deviations as great as or greater than h and k. 

§ (2.) Other Series for the Determination of r. 

For many purposes the series (xix.) is sufficiently convergent to give r for given 
h and k with but few approximations, but we will now turn to other developments. 
We have by (vii.) 



Put X = hf y = k, and write for brevity 

ad ^ be 



c = 



N«HK 
It follows at once from (xix.) that 



(xxL). 






dr 



^^0 



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8 PKOFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 

— qW g-i(it tan « - A«ec «)« ^^ 

if r = sin 6. 

Now either of the quantities under the sign of integration in (xxii.) can be expanded 
in powers of 6 by Maclaurin's theorem. Thus let 

= Xo + (g)/ + (^)|+...+(g)i + ... 
Then 

and it remains to find ( -tx I • 

Now log X = — i (^ tan ^ — A sec dy. 

Hence 

cos* ^ ^ = - X [(A' + ^•*) sin ^ _ M (A - ^ cos 2^)]. 



Differentiating n — 1 times by Leibnitz's theorem, and putting ^ = 0, 
Clearly Xo = ^"**'> then we rapidly find 

Or, finally 

c = ^ + 1M^2 - (A^ + P - A^P) ^ + ;i&{;i.2ifc» - 3(A^ + p) + 5} ^ + . . . (xxiv.), 

where more terms if required can be found by (xxiii.). If 6 be fairly small, 6^ will be 
negligible. Or if A and k be small, the lowest terra in the next factor will be h^ -f /:^, 






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TO THE THEORY OF EVOLUTION. 9 

and this into ^/|5 is generally quite insensible/ Very often two or three terms on the 
right-hand side of (xxiv.) give quite close enough values of ^, and accordingly of 
r = sin^. (xxiv.) is clearly somewhat more convergent than (xix.) if ^ and h are, as 
usually happens, less than unity. 

Returning now to (xix.), let us write it 

e=f{r,h,k). 

This is the equation that must be solved for r. Suppose r^ a root of this when we 
retain only few terms on the right, say a root of the quadratic 

€ = r + p.*r*. 
Then if r = Tq + /»> 

« =/(ro, h, k) + pf{r^, h, k) + \ipY"{r„ h, k) + &c. 



Hence p = ^ ./ vH to a third approximation 



V^-r, 



1 ir:^^;^;^-^;^^ nearly (XXV.), 







which gives us a value of p which, substituted in p^ in the above equation, introduces 
only terms of the 6th order in r,^. 

Another integral expression for c of Equation (xxi.) may here be noticed : 



)v/l 
PutA=-^(^ + y),A: = -^(^-.y) 
Hence 



Jo\/l — ^ 



Jo Vi — T* 

1 — r 
Let tan 2^ = , or, r = co82 if>. 

Therefore 

f46<» 
Jl 1 +V^ 

where t; = cot and is > 1. 

It seems possible that interesting developments for € might be deduced from this 
integral expression. 

VOL. cxcv.— A. o 



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10 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 

§ (3.) To show that the SeiHesfor r is Convergent i/ r < 1, whatever be the Values 

of h and k 

Write the series in the form of p. 6, i.e. : — 
Now 

W,+i = Aw. — WW,_i J •' ^ ' 

From these we deduce 

V»+l = {^' — (2W — 1)} Vn-i — (n — 1) (w — 2) v,_3 
^«+i = {** — (2w — 1)} w,_i — (n — 1) (n — 2) w,_3 

Now let «» = iLi»**|{|»}*, «« =~M'»-i»'*"| {!«}*. 

Then we find 

^'•+* "" v/(n+l)(7. + 2) ^-"^ V 71(71+1) (71 +2; ^'^^'^ ' 

^-■^^ - ^(n + l)(7i + 2) ^-^ V n(7i + l)(ri + 2) ^'-^"^ ' 

Thus, when 7^ is large, we find the ratio of successive terms Sn^Js^ or t^+cj^n is given 

by /o, where 

/) = — 2r — r^//o or, /o = — r. 

The ultimate ratio of s^+2 '*»+2 *^ ^» '« ^s accordingly given by r*, but this is the 
ratio of alternate terms of the original series. The original series thus breaks up 
into two series, one of odd and one of even powers of r. Both these series are 
absolutely convergent whatever h and k be, having an ultimate convergence ratio of r ^ 

§ (4.) To find the Probable Error of the Coirelation Coefficient as Determined by the 

Method of this Memoir. 

Given a division of the total frequency N into a, 6, c, d groups, where 
a + & + c + d = N, then the probable error of any one of them, say a, is '67449 cTa, 
where* 

<r«= y/^^ (xxvi.). 

Let 6 + c? = nj, c + d = 7^2, then 

* The standard deviation of an event which happens np times and fails nq times in n trials is well 
known to be Jnpq. The probable errors here dealt with are throughout, of course, those arising from 
different samples of the same general population. 



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TO THE THEORY OP EVOLUTION. 11 

--. = a/^^^ <r. = ^/^^ . . . (xxvil). 

To obtain Vcd we have, if Br/ denotes an error in any quantity tj, 

Sc + 8d = Sng, 
.-. a-J^ + <r/ + 2a'ca'drcd = <r«.® (xxviii.), 

by squaring, summing for all possible variations in c and rf, and dividing by the total 

number of variations. 

Hence, substituting the values of the standard deviations as foxmd above, we 

deduce 

o-cO-dTed = — ccZ/N (xxix.). 

In a similar manner 

hn^U=hhU + (U)\ 

O'dO'n^rdn^ = O-hO-dTbd + Ct/ 

o-rfo-^^r^^ = d (a + c)/N (xxx.). 

and <rrfO-„,ri/,^= ci (a + &)/N (xxxi.), 

N f* 

Now ^1 ~ "TT"] ^"^^^ 

Thus (r^=NH(rA (xxxiL), 

and similarly Cn^ = NK o-^ . . (xxxiii.). 

Hence the probable error of h 

=^ ^25^13 (^,w.), 

andofi =|^^<I±^±I). ..... .(X.ZV.). 

They can be found at once, therefore, when H and K have been found from an 
ordinate table of the exponential durve, and a, 6, c, d are given. We have thus the 
probable error of the means as found from any double grouping of observations. 

Next, noting that 

8«iK = N*HK8A8Jfc, 
we have (r„, <^njrn,H^ — N^HK Ck o-kr^, 

or Tn^n, = Tkh. 

c 2 



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12 PEOFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 

But 8ni 8% = (S6 + Sd) (8c + 8d), 

= j^ . (xxxyi.), 

therefore 

<Tk(Tkrhk— ygj^ (xxxvil). 

ad — he , ... . 

^^~ v/(T+rf)(aH-c)(c + ei)(a + 6) • • • • • (xxxvui.). 

This is an important result ; it expresses the correlation between errors in the 
position of the means of the two characters under consideration. But if the prob- 
abilities were independent there could be no such correlation. Thus r;^ might be 
taken as a measure of divergence from independent variation. We shall return to 
this point later. 

Since S^i = — HNSA, we have Sn^Sci = — HNSrfS^, whence we easily deduce 

rdn,— --rah (xxxix.). 

Similarly ^if«,= '^'^dk • . (xl.). 

Now d is a function of r, h^ and k. Hence if d = f{r^ A, k)^ 

8rf=^8r + f 8^ + |8ik 

dr dh ok 

= 70^^ + 71^^ + 72^* (xli.)- 

Whence transposing, squaring, summing, and dividing by the total number of 
observations, we find 

yo^o-r^ = cr/ + y^o-^ + y^o-j?' — ^y^a-dO-krdh — ^y^o-dO-kTdk 

+ "^yiy^o-ho-kThk 

Substituting the values of the standard deviations and correlations as found above, 
we have 



V» = j^ { c/(a + 6 + c) + (^^(a + h) {d + c) + (^)'(a + c) (d + b) 



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TO THE THEORY OF EVOLUTION. 13 



It remains now to determine yo, yi, and y^. 
By Equation (i.) 



' _df N 






= -iiJ;l>-"^^ (-!-)> 

where A= ^7-r« - 

Thus 

= '/'2-i (xlv.). 

Sinailarly •/^/(NK) = ^^ — ^ . . .- ^ . . . . (xlvi.). 

H^^^ ^^ = ;7feD"*'^^' ^^=:^D"''^- • • • -(^^^-^^ 

where A= yfJ^ > ^^ = TT^ (xlviii.), 

and thus t/r^ and i/r^ can be found at once from the tables when fi^ and ^82 are found 
from the known values of r, h^ k. 
Lastly, we have from Equation (xxi.) 

d (d + h)(d + e) . 1 f' TJ 
Thus* y^ = d//dr = ~l^U, 

ro/N = xo. 

where Xo = ^ -^i-^ e-.ir^ <»» + **-*") (xlix.) 

a value which can again be foiind as soon as r, ^, A; are known, y^ s ^^ is clearly 
the ordinate of the frequency surface corresponding to x =^ h,y ^= L 
Substituting in Equation (xliiL) we have, after some reductions, 

* By Equations (ii.) and (iii), d + b and d + e are independent of r. 



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14 



PEOFESSOR K. PEAESON ON MATHEMATICAL CONTRIBUTIONS 



Probable error of r s= •67449<rr 

•67449 r (a + d)(e+ b) ^ (a + c)(d + b) ^ (<^ + b)(d+e) 

y/K^ \ 4N« "^ '''» N» "^ '''1 N» 



ad-^hc 'ah — ed , ac — 6d 1 i 



N» 



(1). 



where Xo> ^i> ^^^ ^a ^^^ readily found from Equations (xlix.), (xlvii.), and (xlviii.). 

' Thus the probable error of r can be fairly readily found. It must be noted in using this 
formula, that a is the quadrant in which the mean falls, so that h and k are both 
positive (see fig., p. 2). In other words, we have supposed a + c > 6 + r? and 
a + 6 > c + d. Our lettering must always be arranged so as to suit this result 

j before we apply the above formula. 



§ (5.) To Find a Physical Meaning for the Series in ?% or for the € of Equation (xxL). 

I h 

Return to the original distribution ^ , of p. 2. If the probabilities of the two 
characters or organs were quite independent, we should expect the distribution 



N 



a-\-l a •{- c 

N N 



N 



c •{- d a-i- c 



N 



g -f ft b '\' d 

N N 



N 



c-h db + d 

N N 



N N 

Now re-arranging our actual data we may put it thus 

a -i- b b + d 
a I b 



N N 



N 



N 



e + d a + c ad — be 



N (N 



N 



ad— be 



n: 



N N 



ad --be 



T^T-c + dft + d , orf — 6c 



N 



N 



N 



Accordingly correlation denotes that — jx— has been transferred from each of the 

second and fourth compartments, and the same amount added to each of the first and 
third compartments. If t^ = {ad — 6c)/N^ then ri is the transfer per unit of the total 
frequency. The magnitude of this transfer is clearly a measure of the divergence of 
the statistics from independent variation. It is physically quite as significant as the 
correlation coeflEicient itself, and of course much easier to determine. It must vanish 
with the correlation coefficient. We see from (xxi.) that 

ri = €X HK, 

or we have an interpretation for the series in r of (xix.). 

Now, obviously any function of 17, just like ri itself, would serve as a measure 01 
the divergence from perfectly independent variation. It is convenient to choose a 
function which shall lie arithmetically between and !• 



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TO THE THEORY OF EVOLUTION. 15 

Now consider what happens in the case of perfect correlation, i.e., all the observa- 
tions fall into a straight line. Hence if ad > 6c, either 6 or c is zero, for a straight 
line cannot cut all four compartments, and a and d are obviously positive. Thus c 
and 6 can only be zero if i/ = (c + d){a + c)/N^ or (a + h){b + d)fN\ In order 
that 6 should be zero, it is needful that h and A, as given by (iv.) and (v.), should be 
positive ora+c>6+d, a + 6>c + c?, and the mean fall under the 45° line 
through the vertical and horizontal lines dividing the table into four compartments, 
i.e.y h > k. These conditions would be satisfied if ad >hc and a > d^ c > b. Now 
suppose our four-compartment table arranged so that 

ad>hCy a>d, c>by 
and consider the function 

Q' = ^^^f(a + 6)(5 + cO/N' (^•)' 

or 

^^ . TT ad — be ,,,. V 

Q' = "^"2 (a.f&)(5 + rf) <^->- 

This ftmction vanishes if 17 = 0, and it further = unity if 6 = 0. Thus it agrees 
at the limits and 1 with the value of the correlation coefficient. Again, when h 

and k are both zero, a = d, 6 = c, and Q^ = sin n- ^ . ^ , is thus r by (xx.). Hence 

we have found a ftinction which vanishes with r and equals unity with r, while it is 
also equal to r if the divisions of the table be taken through the medians. 

Now, I take it that these are very good conditions to make for any function ot" 
a, 6, c, d which is to vanish with the " transfer," and to serve as a measure of the 
degree of dependent variability, or what Mr. Yule has termed the degree of 
** association." Mr. Yule has selected for his coefficient of association the expression 

QCUL ""^ OC /« . • • * 

This vanishes with the transfer, equals unity if 6 or c be zero, and minus unity if a 
or (i be zero. The latter is, of course, unnecessary if we agree to arrange a, 6, c, d 
so that ad is always greater than be. Now it is clear that Q2 possesses a great 
advantage over Q^ in rapidity of calculation, but the coefficient of correlation is also 
a coefficient which measures the association, and it is a great advantage to select one 
which agrees to the closest extent with the correlation, for then it enables us to 
determine other important features of the system. 

If we do not make all the above conditions, we easily obtain a number of coeffi- 
cients which woyld vanish with the transfer. Thus for example the correlation Vkjt of 
Equation (xxxviii.) is such an expression.* It has the advantage of a synmietrical 
. form, and has a concise physical meaning. It does not, however, become .imity when 

* In fact (xxxvii) gives us c » a-jiO'tirkk' 



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16 PROFESSOR K, PEARSON ON MATHEMATICAL CONTRIBUTIONS 

either, but not both, 6 and c vanish, nor does it, unless we multiply it by 7r/2 and 
take its sine, equal the coefficient of correlation when a ^=^ d and b = c. 

Again, we might deduce a fairly simple approximation to the coefficient of correla- 
tion from the Equation (xxiv.) for ^, using only its first few terms. Thus we find 

f>*. dd — he /!• \ 

8m 27r ^,^^ _ ^^^, ^ ^^.^^ ^ ^^^^^^^ ^ ^^ (liv.), 

where Xi = V 2 N ' 

X^ "" V 2 N ' 

as an expression which vanishes with the transfer, and will be fairly close to the 
coefficient of correlation. It is not, however, exactly unity when either 6 or c is 
zero. But without entering into a discussion of such expressions, we can write 
several down which fully satisfy the three conditions : — 
(i.) Vanishing with the transfer. 

(ii.) Being equal to unity if 6 or c = 0. 

(iii.) Being equal to the correlation for median divisions. 
Such are, for example : — 

Q« = smjf:j : 26c ^ ,od>}>c . . . (Ivi.), 

{ad — be) (b + c) 



where i^ = 



2 ^1 

Aahcd W 



(ad-'bcy(a-i-d)(b + c) 



Only by actual examination of the numerical results has it seemed possible to pick 
out the most efficient of these coefficients. Q^ was found of little service. The 
following table gives the values of Qg, Q3, Q4, and Q5 in the case of fifteen series 
selected to cover a fairly wide range of values : — 



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TO THE THEORY OF EVOLUTION. 



17 



No. 


r. 


h. 


k. 


Q2. 


Qs. 


04. 


0*. 


1 


•5939 ± -0247 


- •0873 


- ^4163 


•7067 


•6054 


•6168 


•6100 


2 


•5557 ± -0261 


- ^4189 


- 4163 


•6688 


•5657 


•5405 


•5570 


3 


•5529 ± ^0247 


- •0873 


- ^0012 


•6828 


•5809 


•5699 


•5813 


. 4 


•5264 ± -0264 


+ ^2743 


+ •3537 


•6345 


•5331 


•5200 


•5283 


5 


•5213 ± ^0294 


+ ^6413 


+ ^6966 


•6530 


•5511 


•4878 • 


•5160 


6 


•5524 ± -0307 


+ r0234 


+ 3537 


•7130 


•6118 


•6169 


•6138, 


7 


•5422 ± -0288 


+ ^6463 


+ ^5828 


•6693 


•5673 


•5136 


•5452 


8 


•2222 ± •0162 


+ ^3190 


+ ^3190 


•2840 


•2268 


•2164 


•2251 


9 


•3180 ± ^0361 


+ ^1381 


+ ^0696 


•3959 


•3185 


•3176 


•3183 


10 


•5954 ± ^0272 


+ 1^5114 


+ •7414 


•7860 


•7100 


•6099 


•6803 - 


11 


•4708 ± -0292 


+ ^0865 


- -0054 


•5692 


•4712 


•4720 


•4715 


12 


•2335 ± •0335 


+ ^0405 


+ ^0054 


•2996 


•2385 


•2385 


•2385 


13 


•2451 ± ^0205 


+ -2707 


+ ^0873 


•3103 


•2473 


•2456 


•2470 


14 


•1002 ± ^0394 


+ -4557 


+ ^1758 


•1311 


•1032 


•0993 


•1029 


15 


•6928 ± ^0164 


+ -5814 


+ -5814 


•8032 


•7108 


•6699 


•6897 



Now an examination of this table shows that notwithstanding the extreme ele- 
gance and simplicity of Mr. Yule's coefficient of association Q^, the coefficients Q3, 
Qt, and Q5, which satisfy also his requirements, are much nearer to the values 
assumed by the correlation. I take this to be such great gain that it more than 
counterbalances the somewhat greater labour of calculation. If we except cases (6) 
and (10), in which h or k take a large value exceeding unity, we find that Q3, Q^, and 
Qg in the fifteen cases hardly differ by as much as the probable error from the value 
of the correlation. If we take the mean percentage error of the difference between 
the correlation and these coefficients, we find 

Mean difference of Q^ = 24*38 per cent. 
» >> ^3 ^^ 3'9o ,, 

„ Q^= 2-94 
,, Q5= 272 

Thus although there is not much to choose between Q^ and Q5, we can take Q5 as 
a good measure of the degree of independent variation. 

The reader may ask : Why is it needful to seek for such a measure ? Why cannot 
we always use the correlation as determined by the method of this paper ? The 
answer is twofold. We want first to save the labour of calculating r for cases where 
the data are comparatively poor, and so reaching a fairly approximate result rapidly. 
But labour-saving is never a wholly satisfactory excuse for adopting an inferior 
method. The second and chief reason for seeking such a coefficient as Q lies in the 
fact that all our reasoning in this paper is based upon the normality of the frequency. 
We require to free ourselves from this assumption if possible, for the difficulty, as 
is exemplified in Illustration V. below, is to find material which actually obeys 
within the probable errors any such law. Now, by considering the coefficient of 
regression, ra-Jc^ = 8{xy)/Q^a'ia'^), as the slope of the line which best fits the series 

VOL. CXCV, — ^A. D 



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1 8 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 

of points determined as the means of arrays of x for given values of y, we have once 
and for all freed ourselves from the difficulties attendant upon assuming normal 
frequency. We become indifferent to the deviations from that law, merely observing 
how closely or not our means of arrays fall on a line. When we are not given arrays 
but gross grouping under certain divisions, we have seen that the " transfer " is also 
a physical quantity of a significance independent of normality. We want accordingly 
to take a function which vanishes with the transfer, and does not diverge widely 
from the correlation in cases that we can test. Here the correlation is not taken as 
something peculiar to normal distributions, but something significa^t for all distribu- 
tions whatever. Such a function of a suitable kind appears to be given by Qg. 

§ 6. On the " Excess " and its Relation to Correlation and Relative Variability. 

There is another method of dealing with the correlation of characters for which 
we cannot directly discover a quantitative scale which deserves consideration. It 
is capable of fairly wide application, but, unlike the methods previously discussed, it 
requires the data to be collected in a special manner. It has the advantage of not 
applying only to the normal surface of frequency, but to any surface which can be 
converted into a surface of revolution by a slide and two stretches. 

It is well known that not only the normal curve but the normal surface has a 
type form from which all others can be deduced by stretching or stretching and 
sliding. Thus in 1895 the Cambridge Instrument Company made for the instrument 
room at University College, London, a " biprojector," an instrument for giving 
arbitrary stretches in two directions at right angles to any curve. In this manner 
by the use of type-templates we were able to draw a variety of curves with arbi- 
trary parameters, c.gr., all ellipses from one circle, parabolas from one parabola, 
normal curves from one normal curve template. Somewhat later Mr. G. U. Yule 
commenced a model of a normal frequency surface on the Brill system of inter- 
laced curves. This, by the variable amount of slide given to its two rectangular 
systems of normal curves, illustrated the changes from zero to perfect correlation. 
This model was exhibited at a College soirSe in June, 1897. Greometrically this 
property has been taken by Mr. W. F. Sheppard as the basis of his valuable paper 
on correlation in the *Phil. Trans.,' A, vol. 192, pp. 101-167. It is a slight addition 
to, and modification of, his results that I propose to consider in this section. 

The equation to the normal frequency surface is, as we have seen in § 1, 

_ N r /^ 2rxy , y^\ 1 ] 

Now write a;/(<riN/l— r^) = x\ y/c^ = y\ This is merely giving the surface two 
uniform stretches (or squeezes) parallel to the coordinate axes. We have for the 
frequency of pairs lying between x^ x + Sx, and y, 8 + 8//, 



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TO THE THEORY OF EVOLUTION. 19 



Now give the area a uniform slide parallel to the axis of x defined by r/y/l— r* 
at unit distance from that axis. This will not change the basal unit of area 
8a = 8x'By\ and analytically we may write 



Whence we find ^ ^ N ^ , -ti^v 

zoxoy = 2^ 8a expt. (— ^^K*). 

This is the mechanical changing of the Yule-Brill model analytically represented. 
The surface is now one of revolution, and the proof would have been precisely the 
same if we had written in the above results any function/, instead of the expo- 
nential.* It is easy to see that any volume cut off by two planes through the ^is of 
the surface is to the whole volume as the angle between the two planes is to four right 
angles. Further the corresponding volumes of this surface and the original surface 
are to each other as unity to the product of the two stretches. Lastly, any plane 
through the z-axis of the original solid remains a plane through the 2:-axis after the 
two stretches and the slide. These points have all been dealt with by Mr. Sheppard 
(p. 101 et seq.y loc. cit.). I will here adopt his notation r = cosD, and term with him 
D the divergence. Thus cot D is (in the language of the theory of strain) the slide, 
and D is the angle between the strained positions of the original x and y directions. 
Now consider any plane which makes an angle x with the plane of xz before strain. 
Then, since the contour lines of the correlation surface are ellipses, the volumes of 
the surface upon the like shaded opposite angles of the plan diagram below will be 
equal ; and if they be n^ and n^, then n^ + n^ = ^N. If n{ and n^' be the volimies 
after strain, then by what precedes we shall have 

and (rig - ni)/(ni + n^) = « - <)/« + <)• 



-a? 



8 



* The generalisation is not so great as might at first appear, for I have convinced myself that this 
property of conversion into a surface of revolution by stretches and slides does not hold for actual cases 
of markedly skew correlation. 

d2 



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20 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 

Now Ui and n^' will be as the angles between the strained positions of the planes 
bounding n^ and n2* ^^ Aoes not change its direction. Oy is turned through an 
angle 7r/2 — D clockwise, and x becomes x\ say. Hence 

<:<::|-x"+|-I>:l + x"-I+D. 
or (V - n^)/(n^ + n/) = | (x" + D) - 1. 

Let us write E^ = 2(% — n^ and term it the excess for the y-character for the 
line AB. Then we easily find : 

/E, TT , 7r\ . / // . -rw\ cot y' + cot D ,1 ... . 

It remains to determine tan ^ ' and substitute. The stretches alter tan x into 
tan x', such that 

tan X = ~^ tan x- 

Further, by the slide 

cot y" = cot y' — cot D = — ,-^ - -^ cot Y — cot D. 

Hence we have by (Iviii.) above 

- -* (s !) = i.-;/Tr? -t x/(;;7fci oot X oot D - cof D - l) , 

-tan(|jUootD-a55^ (li^). 

\N 2/ (Tg smD ^ ^ 



or. 



Now the excess E^ is the difference of the frequencies in the sum of the strips of 
the volume made by planes parallel to the plane yz on the two sides of the plane AB2; 
(defined by x)> taken without regard to sign. For on one side of the mean yy this is 
ri2 — n^y and on the other — (^i— ^2)- Hence we have this definition of E^, the 
column excess for any line through the mean of a correlation table : Add up the 
frequencies above and below the line in each column and take their differences ivithout 
regard to sign^ and their sum is the column excess. 

If we are dealing with an actual correlation table and not with a method of 
collecting statistics, then care must be taken to properly proportion the frequencies 
in the colunm in which the mean occurs, and also in the groups which are crossed by 
the line. It is the difficulty of doing this satisfactorily, especially if the grouping, as 
in eye and coat colour, is large and somewhat rough, that hinders the effective use of 
the method, if the statistics have not been collected ad hoc. 

Now let E2 be the row excess for the line AB, defined in like manner, then we have 
in the same way 



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TO THE THEORY OP EVOLUTION. 21 

-tan(|^j)=:cotD-^»^ (Ux>). 

\N 2/ a-j^ smD ^ ^ 

Now eliminate o-^/cti between (lix.) and (lix.^") ; then 

(tan (I I) + oot D) (tan (I f ) + cot D) = „^ . 
Whence we deduce 



and, therefore, 



cot D = cot ^^ ~, 

N 2 

DEj + Eg TT ' n \ 
= COS ^ 2 V^-)* 



Substituting for D in (lix.) we find further 

^; = cotxcos(||yco8(|^) (IxL). 

Thus Equations (Ix.) and (Ixi.) give the coefficient of correlation and the relative 
variability of the two characters. The latter is, I believe, quite new, the former novel 
in form. 

If we call nil ^^® frequency in the angle x {-^Ox of the figure above), then it is easy 

to see that E^ = 2(^2 — n^) = N — 471^, and similarly E^ = N — 4m|. Thus 

(El + E2)/N = 2(N — 2(ni + mi))/N. But n^ + m^ is the frequency in the first 

quadrant. This Mr. Sheppard terms P, while that in the second he terms R. We 

have thus (E^ + E2)/N = 2ll/(Il + P), or 

p 
^ = cos^-^7r (Ixii.), 

ie.y Mr. Sheppard's fimdamental result* (* Phil. Trans.,' A, vol. 192, p. 141). 

We can, of course, get Mr. Sheppard's result directly if we put x = 0, when we 
have at once E^ = 2(R — P), Eg = N :;= 2(R + P), and the result follows. 

Equation (Ixi.) may also be written in the form 

^ = cotxsin(^'2^)/8in(^2^) (Ixiii.). 

If we put x^ ^y t^^^ ^1 becomes zero, and the right-hand side of (Ixiii.) is 
indeterminate. If we proceed, however, to the limit by evaluating the frequency in 
an indefinitely thin wedge of angle Xj we reach merely the identity cja-^ = o-i/o-^. 
Hence there is no result corresponding to (Ixi.) to be obtained by taking 
Mr. Sheppard's case of x = 0. 

The following are the values of the probable errors of the quantities involved : — 

* In the actual classification of data (Ix.) and (Ixii.) suggest quite different processes. We can apply (Ix.) 
wbere (Ixii.) is difficult or impossible, e.g., correlation in shading of birds' eggs from the same clutch. 



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22 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS ^'^'^^ r 

Probable error of E^ = -67449 x/N (1 - Ei*/N») .... {\xvt.)\ 
E^ = -67449 n/N(1 - E^/N*) .... (Ixv.). ^ 

Correlation between errors in E. and E, = - a/ J! " "^^(S! i! T & S-i • • (l^vL). 

^ * V (1 + i-i/N) (I + Ej/N) ^ ' 

, Tj , , , . -67449 sin D*/D(^:^D) • /, •• x 

Probable error m r = ^ r .... (Irvii.), 

where D = ' * ^ {cf. Sheppard, loc. cit., p. 148). 

Probable error in ratio Ci/o-j = 

•67449 o-^TTf/, E,«\. oi%A L fn E,2\. 2./E,^\ 

The application of the method here discussed to statistics without quantitative 
scale can now be indicated. If the characters we are dealing with have the same 
scale, although it be unknown, then, if the quantitative order be maintained, i.e., 
individuals arranged in order of lightness or darkness of coat or eye-colour, the 
diagonal line on the table at 45° will remain unchanged, however we may suppose 
parts of the scale to be distorted, for the distortion will be the same at corresponding 
points of both axes. Further, if we suppose the mean of the two characters to be the 
same, this 45° line will pass through that mean, and will serve for the line AB of the 
above investigation. In this case we must take tan ^ = 1, and consequently (Ixi.) 
becomes 

iTj(r.2 = cos y f )/cos (^fj (Ixix.). 

We can even, when the mean is a considerable way off the 45° line, get, in some 
cases, good results. Thus, the correlation in stature of husband and wife worked out 
by the ordinary product moment process is '2872. But in this case E^ = 382*062 
Ej = 806'425, and this gives the correlation '2994. On the other hand, the actual 
ratio of variabilities is 1*12, while (Ixix.) makes it 276 ! This arises from the fact 
that the errors in E^ and E2, due to the mean being off the 45° line, tend to cancel in 
El + E^, but tend in directly opposite directions in the ratio of the cosines. Similarly 
the correlation between father and son works out *5666, which may be compared with 
the values given in Illustration V. below, ranging from '5198 to '5939. Again, 
correlation in eye-colour between husband and wife came out by the excess process 
•0986, and by the process given earlier in the present Memoir '1002. But all these are 
favourable examples, and many others gave much worse results. We ought really only 
to apply it to find cja-^ when the means are on the 45° line, as in the correlation of the 



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TO THE THEORY OF EVOLUTION. 



23 



same character in brethren, and even in this case the statistics ought to be collected 
ad hoc, i.e., we ought to make a very full quantitative order, and then notice for each 
individual case the number above and below the type. For example, suppose we had 
a diagram of some twenty-five to thirty eye tints in order {e.g., like Bertrand's), 
then we take any individual, note his tint, and observe how many relatives of a 
particular class — ^brethren or cousins, say — have lighter and how many darker 
eyes ; the difference of the two would be the excess for this individual. The same 
plan would be possible with horses' coat-colour and other characters. After trying the 
plan of the excesses on the data at my disposal for horses' coat-colour and human eye- 
colour (which were not collected ad hoc), I abandoned it for the earlier method of 
this Memoir ; for, the classification being in large groups, the proportioning of the 
excess (as well as the differences in the means) introduced too great errors for such 
investigations. 



§ 7. On a Generalisation of the Fundamental Theorenx of the Present Memoir. 

If we measure deviations in units of standard deviations, we may take for the 
equation to the correlation surface for n variables 

N 



z = 



(27r)VE 



e-^W^^)-H^)} (Ixx.), 



where 



K = 



1 

Til 



1 



'23. 






^n-1, 1 



^,2 






'»-i, » 



rn- 



n— 1, n 



and 'Rpg is the minor obtained by striking out the ^th row and qth column, rpg is, of 
course, the correlation between the pth and ^th variables, and equals r^. S^ denotes 
a summation for s from 1 to n, and Sg a summation of every possible pair out of the 
n quantities 1 to n. 

Now take the logarithmic differential of z with regard to r^ We find 



For 



dB,/dr^ = 2Kp^ 



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24 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 

and, generally, whether 5 is or is not = s, or these are or are not = p and g, we 
have 

dir[-R)= n* O^xi.). 



This follows thus : 



dr„ \ K / R drp, R2 dvp^ " R 1^ 



R2 ' 



or we have to show that 



djxggt £t]\„i \xpq — Kj»t R^#/ — Rpi* Ry* 
dvpq "" R 

R«l^lVg — RyjRyjf , R<^Ryy — fiw/ Ryj 

~ R "*" R 

where jb^R«/ is the minor corresponding to the term r^q in R,^, and q^s$> the minor 
corresponding to the term r^p.* But this last result is obvious because R^^ only con- 
tains Vpq in two places, i.e.y as Vp^ and r^p. 

Putting s =s\ we have the other identity required above, i.e.^ 



drpqXHJ R9 



Returning now to the value for - - — on the previous page, we see that the two 



z dvpq 



sum terms may be expressed as a product, or we may put 

-4=-^+«.(l-)xs.(l-) 

Now write 



(2Tr)'VK' 



z =- .„ ... /:rt e-*. 



Tx 1 dz d^(h . d(b d(b 

Hence - -— = — — ^ + ^ -:^ 

z dVpg dxp dj:^ dxp dxg 

Now differentiate log z with regard to x^. Then 

dz d<l> 

dosp dXp 

* See also Soott, * Theory of Detenninante,' p. 59, 



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TO THE THEORY OF EVOLUTION. 25 



Thus finally 





d,uyixj 


= — 




dz d4> 
djCg do:p 


1 

z 


(Pz 


dz 


^4> , 

dX^g 
dh 


d^d^ 
dxp dxg 






dr„ 


dayfajj • 





(Ixxii.). 



In other words, the operator djdr^ acting on z can always be replaced by the 
operator d^jdxpdxq. 

Let d/dppq denote the effect of applying the operator d/dvp^ to «, and putting Vpg 
zero after all differentiations have been performed, then the effect of this operator will 
be the same as if we used d^jdxpdxq on 2;, putting Vpq zero before differentiation. 
Generally, let F be any series of operations like d/dvp^^ then we see that 



\drpq' dryq, ' 



ipq Uipig, dVp..^,, 



\ dXj/d.Cq dxpdxq, ' d^Vp,dXg„ ' / (27r)** 



(x.») 



Now let F be the function which gives the operation of expanding z by Maclaurin's 
theorem in powers of the correlation coefficients, i.e., 

F = e««^-'5^)> 
then 

z = e ^(-' .7^^ z = — M-' 1^) e - »«'<'-^ . 

This is the generalised form of result (xiv.) reached above. 

Now let Zr. — ■ ^ • ^-iSi<''"> 

nov. iGT, Zq — ^2^^^^ € , 

then z^^ is the ordinate of a frequency surface of the nth order, in which the distribution 
of the n variables is absolutely independent. We have accordingly the extremely 
interesting geometrical interpretation that the operator 

applied to a surface of frequency for n independent variables converts it into a surface 
of frequency for n dependent variables, the correlation between the sth and sth 
variables being r„,* 

^ I should like to suggest to the pure mathematician the interest which a study of such operators would 
have, and in particular of the generalised form of projection in hyperspace indicated by them. 
VOL. CXCV. — A. E 



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26 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 

Expanding, we have 

+ • • • +||{Sa(»'«'d^,)}"^o + . •. . ■ . . (Ixxiii.). 
Our next stage is to evaluate the operation 

Hj'*''d^ Co- 
llet us put 

,Vi — x„ ,«8 = a?/ — 1, ,V3 = a;,(a;/ — 3), 

and ,Vp = the pth function of «, as defined by (xv.). 

Let €, be a symbol such that c/ represents ,Vp. Then we shall show that 

^^"•^.j =^o = «o{s,(r,^^^)}"' (Ixxiv.). 

We shall prove this by induction. 
By (xii.) 

,v^+i = X, ,Vm — m ,t;«.i, 

or c/"*"^ = X, c/ — m c/""\ 

and by (xvi.) 

^ = m,i;«_,. or — = m e,- ^ 
Now, let X (€*) be any function of c. 



if we suppose it can be expanded in powers of c, 
Then 

d 



= S(A,<?e,»-i) 

= S(A,(x,€,»-e/+i)) 

= a!^(A, €,1) - €. S(A, €/) 

= («*-«') X(«') (Ixxv.). 

Similarly ;^;;j^ x(«* .«'') = (^'-«')(^'—«'') x(«« «*') • • • • (Ixxvi.). 

Now suppose that 



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TO THE THEORY OF EVOLUTION. 27 

then 



{^«(^-d^)}'""^o = S,(^-^>oU, 



where U stands for. {S2(r,^€,€^)}*. 

Hence, remembering that dzjdx^ = — z^t>y 



= «oS»(»'«.<»e^)U' 
= 2o{S2(r^€^^)}-+', 

which had to be proved. 

But it is easy to show by simple differentiation that 

^ Zo {^i{r^^,)Y (Ixxvii.). 

Hence the theorem is generally true. 
Thus we conclude that 

+ • • • + |4 {S2(^'^'*.)}'"+ . • • •] O^^viii.). 

It is quite straightforward, if laborious, to write down the expansion for any number 
of variables. 

Now let Q be the total frequency of complices of variables with x^ lying between 
h^ and 00 , X2 between h^ and oo , . . . aj, between h, and oo , . . . 5c„ between h^ and oo ; 
and let Qq be the frequency of such complices if there were no correlations. 
Then 



I ... I ... 2; dx^ dx^ . . . dxt 



dx^ 






fCD •flO ttOO ftO 

I ... I ... 1 ZQdx^dxci . . . dx, . . . dXf^ 
hi JA, Ja. Ja. 

Now let 

£ 2 



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28 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 

where 

We have Qo = 'iHfiA . . A • • i8« H^ Hg . . H, . . H,. 

But by (xviii.) 



where 

and as above, 

Thus 



,W/,_l = [,Uj,_l]^ = ». (Ixxx.), 

^, = ^'e-i^dx./e-i'"\ (Ixxxi.). 



(vfel" 1 • • • 1 • ' * L '^^'' V'-'V- • • e-*<"'+'^+ . . . +x^+ . . . ^-.^dxidx^. ..dx,...dx„ 
= H,H, . . . H, . . . H^A . . . i8, . . . /8/%i4=i4^ . . . , 

Ps Pr Pt" 



or 



j j . . . j . . . j ZqU{,v^) dx^dxc^ . . . dx, . . . dxn = QqH r^) 

. . . . (Ixxxii.). 

where 11 denotes a product of ,Vj, for any number of v's with any s and p. The rule, 
therefore, is very simple. We must expand the value of z ini/& as given by (Ixxviii.) 
above, then the multiple integral of this will be obtained by lowering every v's right- 
hand subscript by unity (remembering that sVq =1), and further dividing by the fi of 
the left-hand subscript. The general expression up to terms of the fourth order has 
been written down ; it involves thirty-four siuns, each represented by a type term 
All these would only occur in the case of the correlation of eight organs, or when we 
have to deal with twenty-eight coefficients of correlation. Such a number seems 
beyond our present power of arithmetical manipulation, so that T have not printed the 
general expressions. At the same time, the theory of multiple correlation is of such 
great importance for problems of evolution, in which over and over again we have 
three or four correlated characters to deal with,'*^ that it seems desirable to place 
on record the expansion for these cases. I give four variables up to the fourth and 
three variables up to the fifth order terms. Afterwards I will consider special cases. 



* In my memoir on Prehistoric Stature I have dealt with five correlated organs, *.«., ten coefficients. In 
some barometric investigations now in hand we propose to deal with at least fifteen coefficients, while 
Mr. Bramley-Moore, in the correlation of parts of the skeleton, has, in a memoir not yet published, dealt 
with between forty and fifty cases of four variables or six coefficients. 



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TO THE THEORY OF EVOLUTION. 29 



Vcdue of the Quadruple Integral in the Case of Four Variables.^ 

Qo fil^% A^3 ^1^4 ^3^3 ^2^4 ^8^4 

^1^3^8 ^ ^1^2^3 ^ A^2^3^4 A^2^3^4 



^^18^*84 //' I ^!li^J4 ,, iv _j_ ^^'28^'8. 



^ ^^18^^84 ^ /// _^ fhj^u t; >^ + fVM <y ' 

+ 



^^*24^84 «i iv I ^^12^*24 «. " ^_ ^^18^*24 
^2^3^4 ^ AA^4 ^ ^i^2^3^4 

i 2^14^24 iv I ^^28^24 //I 

+/3i/32/34 ''^ + fiJ3^, ""' J 
+ 1^1/31/8/^''^ ^ Ms'"'''' +/8i/8/^^^ +/8a''^'^ +/92/8/^^^ 
+ /3A '* ''^ + M^ '''''' + Ay8A ''^''^ + /^^ ''^''^ 

^^l^^2^>3 P1P2P8 P1P2P3 

3^13?^ ,„ „ 3r^/ro3 , .^, ^^'u^'23^ " /// 

+ A/8^A ''^ "^ + /9i/9^. ''^ "' + A/3A "^ ''' 

* To simplify the notation, »,', »,", «;,'" »,'» have been used for i»„ jf,, 8»«, 4»»- 



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30 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 

+ /8A/9A ^ ^ +/3i/S,/93^4 ^ > +/8i/8^A ^^ 
6rigrggr34 , ,„ 6r)3r33^^^ ,„ 6ri4rg8r^ ,„ .^ 

+ )8,^A ^ ^ ^ + /3i;SA ''^''^ "^^ J 
+ liW, " * +i8ii9s '''» +A/8/'' ' +/3W38 * ' +.8^4'*^ ^/Sa'^''^^ 

^ i8i/8^8 "^^ '■' ^ iSi^A "^^ ' + ^A^4 "^^ ' ^ /e./8A/84 ''^ ''« 
+ /Sii8,i94 * '^ + i8>^A ' '^ + /8i/3w38^« "^'"^^ + fi^M^. '' ^' 
+ /3i/9A ^^ + i8i/8A ^ '^ ^ /3^A " * ^ /9,/88i8. " * 

, ^^'l8 ^34 / // , ^ 4?j£s/_ „, j^ ^^'l8^ ^SA /// / i ^^'l8^84^ /// jy 

, ^14' ^84 / iy , ^^uV „ '// w , 4rj/»^ „ „ 4rg3»^ ,„ j^ 
+ /Si/Ss-S, ^«^« + A/8A ' ' + /8^A ' ' ^ M^* ' ' 

+ ^AA "' "' + ^. "' "• + ^ "• ' 



. 6r„ 



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TO THE THEORY OF EVOLUTION. 31 

I -^--^^'la ^14^*24 f ft \v % -*■ ^^12^14 ^'24 / // iv I "'••^^12^14^91 / // ;» 

+ A)8A ^'^^^ ^» + A/8A ^ ' -^ ^ A/S3/S, ^'»^* ^« 



IV 



+ A/SW3A ^ » * + /3i/8^A ' ' + /3i/3^A ^ ' ^ » 



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32 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 

^ /Ss/8A '''' ""^ ""' + /8^,/3, *^ ""' ''« + /9^,/3, ''l ''^ ''^ 

, 2^12 ^18^14^23 / // m , 247-13 ^13^1^ , „ ;^ 24rigri3rayi4 / ^/ //' 
^ /8i/32)8A "^^"^^"^^ + /8,/82/33/8, ^'^^^ ^^ ^- ^1/8,^3/3, ^^ "« '^^ 

+ i9ii82/9A ^ '^ ^ + /9i)82/9A ^1^1^^ ^1 + /81/3,^3/S, ^«^i ^1 

+ /3i;8,^3/3, ^^^i"« + /9i/33)S3/3, ^^^^^1 ^1 + /3i;8,/33/9, ^'^ ^* ^^ 

+ A/S^a^* ""^'^ ""' + Ay92/9A ^^^1^^ + fi^^^, ^1^1 ^'^ 

. . . . (Ixxxiii.). 

In the case of three variables, we must cancel in the above expression all terms 
involving jS.^ Thus we shall have 3 instead of 6 first order terms, 6 instead of 21 
second order terms, 10 instead of 56 third order terms, and 15 instead of 126 fourth 
order terms — a much more manageable series. 

I give below the extra term necessary for calculating the value of (Q — Qo)/Qo a.s 
far as the fifth order terms in the case of three variables. 



Fifth Order Terms for Three Vanables. 

PiPiP% PiPA PiPoPa 

ins** iHtS*"^ lO'S'*'* 



+ 



20r. 






f/t 



+ ^^A'"^-^^''^« + /3./3A '^^^''^ I ^^''''''^'^•^• 



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TO THE THEORY OF EVOLUTION. 33 

A numerical illustration of these formulsB will be given in the latter part of this 
Memoir. It will, however, be clear that what we want are tables of log ( ^ j, including 

log ('^) or log [—1 for a series of values of h. Such tables would render the compu- 
tation of — jr-^ fairly direct and rapid ; they could be fairly easily calculated from 

existing tables for the ordinate and area of the normal curve, and I hope later to 
find some one willing to undertake them. 

Meanwhile let us look at special cases. In the first place, suppose, in the case of 
three variables, that the division of the groups is taken at the mean, ?.e., ^^ = A^ = 
A3 = 0. Then we have 



/8i = A = )83 = fV*''<^=»=A/|- 
v/ = v{ = v/" = 

< = < = <'= -1 

< = v^' = <' = 

< = < = <" = 3. 

Hence we have 

= Qojl + — (sin-irij + sin-'ris + sin-^rja)! (Ixxxv.). 

Let ri2 = cos D^, rjg = cos D^g, r^ = cos Djg, and let E be the spherical excess of 
the spherical triangle whose angles are the divergences D^, Djj, D23. Then 
we have 



Q — Qo ^ _ ?E T\ r» r\ —.'^ 
Qo 2 - 2 ~ ^i» ~ ^^3 ~ ^«8 - 2 



Or: sin^-^J=cosE (Ixxxvi.). 

Now take the case of four variables. Here we have 

^1 = ^2 = ^3 = ^4 = V 5 

< = V = <" = 1Jji» = 1 

and all the odd v's zero. Hence 

VOL. cxcv. — ^A. p 



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34 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 

Q 2' / 2 \® 

— Q— ^ = ;. ('-12 + ''js + »*it + ^23 + »'24 + r^) + [-) ('•w'as + rijra, + r^^r^^) 

*> 1 / 2 \® 

+ ^ ]3 (**•«' + ''is" + '■"' + ''2S' + ^2*' + **»*') + (^ j (*'18'*13'*14 + ^12* 

/2 V 1 

+ '-Isna'M + »*]*''24»*84) + \^y i3"(^l*'*'28 + '•l4»'23' + ^3^24* + ^3% + ^-llV,^ 

/ 2 \^ 1 

+ ^12^3*^) + [yj 12" (^12^^14^23 + ^Au'^SS + ^12^''l3^54 + ^18^'u^^24 + ^13^23^^24 

+ ^4^23^ + ^2^13^4 + ^12^^34 + ^2^23^^*34 + ^14^28^34^ + ^2^24^^^ 

+ ^13^24^34^) + (IxXXvii.). 

This is the correct value including terms of the fourth order, but to this order of 
approximation we can throw it into a much simpler form. Let r,y = sin 8,^, then 

— p-— ^ ^ = sin"^ rjg + sin"^ rjg + sin~^ r^^ + sin"^ Vc^ + sin"^ r^^ + sin~^ r^ 

2 . 
+ — (sin"^ 7*13 sin""^ r^^ sin""^ r^^ + sin"^ r^^ sin"^ 7\2Z ^^^"^ ^24 

+ sin~^ r^g sin~^ r^^ sin""^ r^ + sin""^ r^^ sin"^ r^^ sin"^ r^^) 

+ ~ [sin-» n* sin-^ r«,{(l - r,,«) (1 - n,') (1 - r^») (1 - r,,*)]-* 
+ sin-i ri^sin-i r3,{(l - r,,^) (1 - r,,*) (1 - r^') (1 - V)}"* 
+ sin-i r,3 sin-i r,,{(l - V) (1 - r,,«) (1 - V) (1 - rg/)}"*] 
= 8j3 + 8i3 + 8j4 + 823 + 824 + ^34 

+ ^ (812813^14 + 813833824 + 813823834 + 814824834) 

2^ A^ggg COS 8|^ COS 8g3 4- 8128^ COB 812 COS 83^ 4- SigSg^cofr 8,^00382 A 

^ \ COS 813 COS Si3 cos Si4 COS Sjj cos 834 COS Sjj4 / 

. . . • (Ixxxviii.). 

We may write this sin — ^— ^ ^ = cosE' (Ixxxix.) 

where 

E' = "2 "■ 812 — 813 — 814 — 823 — 824 — 834 

— ^ (812833814 + 810823834 + 813823824 + 814824834) 

2 / 8i^833 cos 834 cos 823 -f 8,38^005812008834 4- 813824 cos 8i3 COB 83t \ 
IT \ COB 8j3 COS Si3 COS 834 COS 853 COS 834 COS 834 / 

The expressions E and E' of (Ixxxvi.) and (Ixxxix.) are of considerable interest, for 
they enable us to express the area of a spherical triangle in three-dimensioned space, 



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35 



and (up to the above degree of approximation) the volume of a " tetrahedron " on a 
" sphere " in hyperspace of four dimensions. In fact, the whole theory of hyperspace 
" spherical trigonometry " needs investigation in relation to the properties of multiple 
correlation. 

In our illustrations (viii.) and (ix.) will be found examples of the above formulae 
applied to important cases in triple and quadruple correlation in the theory 
of heredity. I consider that the formulae above given will cover numerous novel 
applications, for many of which greater simplicity will be introduced owing to the 
choice of special values for the h's or for the correlation coefficients. 

(8.) Illustrations of the New Methods. 

Illustration I. Inheritance of Coat- colour in Horses. — The following represents 
the distribution of sires and fillies in 1050 cases of thoroughbred racehorses, the 
grouping being made into aU coat-colour classed as ** bay and darker," " chesnut and 
lighter":— 





Colour. 


Sires. 




Bay and 
darker. 


Chesnut and 
lighter. 


FiUiea. 


Bay and darker . . . 


631 


125 


756 


Chesnut and lighter . 


147 


147 


294 






778 


272 


1050 



a 


h 


a + b 


c 


d 


c + d 


a + c 


b + d 


N 



Then we require the correlation between sire and filly in the matter of coat-colour, 
and also the probable error of its determination. 
We have from (iv.) and (v.) 

«i = ^ \^ -^ = \/- \e-^^dx = •481.905, 

«2 = ^ V^ ^ = V ^Jo^ '^^y = -440,000. 

Hence from the probability integral tables 

h = -64630, k = -58284. 

We have then : log HK = T-037,3514 by (xvii.), 

F 2 



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36 PROFESSOE K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 

Thence c = "'~J = -619,068 from (xxi.). 

Calculating out the coefficients of the series in r in (xix.) we find 

•619,068 = r + •188,345r2 + •064,0814r« + •107,8220r* + '005,99867^ + •067,2682r« 

4- &c. 

Neglecting powers of r above the second, we find by solving the quadratic and 

taking the positive root 

r — -5600. 

Solving by two approximations the sextic we finally determine 

. r = -5422, 

correct, I think, to four places of figures. 

Turning now to the probable error as given by Equation (1.), I find 

^s + jfc2 _ 2rhk = -348,924, 
and from (xlix.) 

logxo= 1*170,0947. 

Further : '^ll\^ ~ -275,642 , Jpi^ = '393,078. 

Hence from (xlvii.) and (xlviii.) we find 

1 r-»93,078 1 r -275,642 

and by means of the probability integral table 

i|ri = -108,884, xji^ = -152,865. 

By substituting in (1.), we find 

probable error of r = '0288. 
From (xxxiv.) and (xxxv.) we find 

p.e. of A = -0282. p.e. oi k = -0278. 

Thus, finally, we may sum up our results 

h = -6463 ± -0282, k = '5828 ± '0278, 

r = -5422 ± -0288. 

The probable en-or of this r, if we had been able to find it from the product 
moment, would have been '0147, or only about one-half its present value. 



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TO THE THEORY OF EVOLUTION. 37 

Illustration IL — Our analysis opens a large field suggested by the following 
problem : — What is the chance that an exceptional man is born of an exceptional 
father f 

Of course much depends on how we define " exceptional," and any numerical 
measure of it must be quite arbitrary. As an illustration, let us take a man who 
possesses a character only possessed by one man in twenty as exceptional. For 
example, only one man in twenty is more than 6 feet 1'2 inches in height, and such a 
stature may be considered " exceptional." In a class of twenty students we generally 
find one of " exceptional " ability, and so on. Accordingly we have classed fathers and 
sons who possess characters only possessed by one man in twenty as exceptional. We 
first determine h and Jc, so that the tail of the frequency curve cut off is -^ of its 
whole area. This gives na h = k = 1*64485. 

Next we determine HK = ^e'^'''^'*\ and find log HK = 2-026,8228. 

Then we calculate the coefficients of the various powers of 7* in (xix.). We find 

\og^hk= -131,2225. 

logi(/t« - 1) (F - 1) = 1-685,5683. 

log^Ah^ - 3)(^ - 3) = 3-990,1176. 

logxio (A* - 6/i* + 3)(** - 6F + 3) = 1-464,4772. 

log^(A* - 10A« + 15)(** - lOP + 15) = 2-9.25,6367. 

It remains to determine what value we shall give to r, the paternal correlation. It 
ranges from "3 to "5 for my own measurements as we turn from blended to exclusive 
inheritance. Taking these two extreme values we find 

— ^r=r- = -0046344 or '0096779. 

But — 2 = — — ^ :^ -^ and the second term is the chance of exceptional 

fathers with exceptional sons, when variation is independent, i*e., when there is no 
heredity, = ^ X Vo = '0025. 

Thus d/N = -007134 or -012178 ; 

accordingly 6/N = -042866 or -037822. 

Hence we conclude that of the 5 per cent, of exceptional men -71 per cent, in the 
first case, and 1*22 per cent, in the second case, are bom of exceptional fathers, and 
4-29 per cent, in the first case and 3-78 per cent, in the second case of non-exceptional 
lathers. In other words, out of 1000 men of mark we may expect 142 in the first case. 



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38 



PEOFESSOR K. PEAESON ON MATHEMATICAL CONTRIBUTIONS 



244 in the second, to be born of exceptional parents, while 858 in the first and 756 in 
the second are born of undistinguished fathers. In the former case the odds are about 
6 to 1, in the latter 3 to 1 against a distinguished son having a distinguished father. 
This result confirms what I have elsewhere stated, that we trust to the great mass of 
our population for the bulk of our distinguished men. On the other hand it does not 
invalidate what I have written on the importance of creating good stock, for a good 
stock means a bias largely above that due to an exceptional father alone. 

In addition to this the •^- of the population forming the exceptional fathers pro- 
duce 142 or 244 exceptional sons to compare with the 858 or 756 exceptional sons 
produced by the ^| of the population who are non-exceptional. That is to say that 
the relative production is as 142 to 45*2, or 244 to 39*8, i.e., in the one case as more 
than 3 to 1, in the other case as more than 6 to 1. In other wordsy exceptional 
fathers produce exceptional sons at a rate 3 to 6 times as great as non-exceptional 
fathers. It is only because exceptional fathers are themselves so rare that we must 
trust for the bulk of our distinguished men to the non-exceptional class. 

Illustration III. Heredity in Coat-colour of Hounds. — To find the correlation 
in coat-colour between Basset hounds which are half-brethren, say, ofispring of the 
same dam. 

Here the classification is simply into lemon and white ijiiv) and lemon, black and 
white or tricolour (f), 

The following is the table for 4172 cases : — 



Colour. 


t. 


IV). 


Totals. 


t. 


1766 


842 


2608 


Iw. 


842 


722 


1564 


Totals 


2608 


1564 


4172 



Proceeding precisely in the same way as in the first illustration we find 



«! = Oj = 
h=zh = 



25024 

318,957 

157,6378 



logKH=:l- 

c= -226,234. 

It will be suflficient now to go to r*. We have 

•226,234 = r + '050,867 r* + -134,480 r» + '035,587 v\ 



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39 



The quadratic gives r = '2237. Using the Newtonian method of approximating to 

the root we find 

r = -2222. 

Summing up as before, after finding the probable errors, we have 

h = k= -3190 ± -0133, 
r= -2222 ±-0162. 

Illustration IV, Inheritance of Eye-colotir in Man.— To find the correlation in 
eye-colour between a maternal grandmother and her granddaughter. Here the 
classification is into eyes described as grey or lighter, and eyes described as dark grey 
or darker.* 





Tint. 


Maternal grandmother. 


Totals. 


Grey or lighter. 


Dark grey or 
darker. 


1 

1 

1 

C5 


Grey or lighter .... 


254 


136 


390 


Dark grey or darker . , 


156 


193 


349 


Totals 


410 


329 


739 



As before, we find 

aj = -109,607, ajj = -055,480, 

h = -138,105, k = -069,593, 

log HK = T-196,6267, 
c = -323,760. 
Series for r up to r* 

•323,760 = r + -004,806?^ + -162-696r5 + '000,358^* 

The quadratic gives r = "3233, and the biquadratic 

r = -3180, 

the value of the term in r* being -000,00366, so that higher terms may be neglected. 
Determining the probable errors as in Illustration I., we sum up : — 

* According to Mr. Galton's classification, the first group contains eyes described as light blue, blue, 
dark blue, blue-green, grey ; and the second eyes described as dark grey, hazel, light brown, brown, dark 
brown, verv dark brown, black. 



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40 PKOFESSOE K. PEAESON ON MATHEMATICAL CONTBIBUTIONS 

h = -1381 ± 0312, 
k = -0696 db 'OSll, 
r = -3180 ± -0361. 

Illustration V. Inheritance of Stature. — The following data have been found for 
the inheritance of stature between father and son from my Family Data cards, 1078 
cases : — 

Mean stature of father. . . . 67"-698 

son 68''-661 

Standard deviation of father . . 2"7048 

son. . . 2"7321 

Correlation = -5198 ± '0150. 

Now for purposes of comparison of methods the correlation has been determined 
for this material from various groupings of fathers and sons : — 



(A.) 



Fathers. 



« 



Class. 


Below 67"-5. 


Above 67"-5. 


Totals. 


Below 67"-6 . . 


269-25 


96-75 


365 


Above 67"-5. . 


232-25 


480-75 


713 


Totak . . . 


501-6 


576-5 


1078 



(B.) 



Fathers, 



CQ 



Class. 


Below 66"-5. 


Above 66"-5. 


Totals. 


Below 67"-5 . . 


211-25 


153-75 


365 


Above 67"-5 . . 


152-75 


560-25 


713 


Totals . . . 


364 


714 


1078 



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TO THE THEORY OF EVOLUTION. 
Fathers, 



41 



Class. 


Below 67"-5. 


Above 67"-5. 


Totals. 


Below 68-6" . . 


356-26 


182-25 


638-5 


Above 68'5" . . 


146-25 


394-25 


539-6 


Totals . . . 


501-6 


676-6 


1078 



(D.) 



Fathers. 



CQ 



CSass. 


Below 68"-6. 


Above 68"-5. 


Totals. 


Below 69"-6 , . 


506 


182 


688 


Above 69"-5 . . 


149-5 


240-6 


390 


Totals . . . 


655-6 


422-6 


1078 



(E.) 



Fathers. 



Class. 


Below 69"-5. 


Above 69"-6. 


Totals. 


Below 70"-5 . . 


669 


147 


816 


Above 70"-6 . . 


128 


134 


262 


Totals . . . 


797 


281 


1078 



(F.) 



« 



Fathers, 



Class. 


Below 70"-5. 


Above 70"-5. 


Totals. 


Below 69"-5 . . 


641-25 


46-76 


688 


Above 69"-5 . . 


271-75 


118-25 


390 


Totals . . . 


913 


165 


1078 



VOL. OXCV. — A. 



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PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 



Table of Results. 



Claasificatiom. 


Correlation. 


Mean of sons. 


Mean of fathers. 


A 
B 

D 
E 
F 


•5939 ± -0247 
■5557 ± -0261 
•5529 ± -0247 
-5264 ± -0264 
•5213 ± 0294 
-5524 ± -0307 


k. 
68"-64(- -416,32) 
68"-64(- -416,32) 
68"-50(- -001,16) 
68"-53 (-353,71) 
68"-60 (-696,57) 
68"-63 (-353,71) 


A. 
67"-74(- -087,00) 
67"-63(- -418,86) 
67"-74(- -087,30) 
67"-77 (-274,30) 
67"-76 (-641,30) 
67"-73 (1-023,44) 



Now these results are of quite peculiar interest They show us : — 

(i.) That the probable error of r, as found by the present method, increases with 
h and k But the increase is not very rapid, so that the probable errors of the series 
range only between '025 and •031. Hence while it is an advantage, it is not a very 
great advantage, to take the divisions of the groups near the medians. It is an 
advantage which may be easily counterbalanced by some practical gain in the method 
of observation when the division is not close to the medians. 

(ii.) While the probable error, as found from the present method of calculation, is 
1*5 to 2 times the probable error as found from the product moment, it is by no 
means so large as to seriously weigh against the new process, if the old is un- 
available. It is quite true that the results given by the present process for six 
arbitrary divisions diflPer very considerably among themselves. But a consideration 
, of the probable errors shows that the differences are sensibly larger than the prob- 
; able error of the differences, even in some case double ; hence it is not the method 
, but the assumption of normal correlation for such distributions which is at fault. As 
' we shall hardly get a better variable than stature to hypothesise normality for, we 
see the weakness of the position which assumes without qualification the generality 
of the Gaussian law of frequency. 

(iii.) We cannot assert that the smaller the probable error the more nearly will 
the correlation, as given by the present process, agree with its value as found by 
the product moment. If we did we should discard '5213, a very accordant result, 
in favour of '5529, or even '5939. The fact is that the higher the correlation the 
lower, ceteris "paribus^ the probable error, and this fact may obscure the really best 
result. Judging by the smallness of h and k and of the probable error, we should 
be inclined to select C or the value '5529. This only differs from '5198 by slightly 
more than the probable error of the difference ('033 as compared with '029) ; but 
since both are found from the same statistics, and not from different samplings ot 
the same population, this forms sufficient evidence in itself of want of normality. 
The approximate character of all results based on the theory of normal frequency 
must be carefully borne in mind ; and all we ought to conclude from the present 



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data for inheritance of stature from father to son would be that the correlation 
= '55 ± '015, while the product moment method would tell us more definitely that 
its value was '52 ± -015. There is no question that the latter method is the better, 
but this does not hinder the new method from being extremely serviceable; for 
many cases it is the only one available. 

Illustration VI. Effectiveness of Vaccination. — To find the correlation between 
strength to resist small-pox and the degree of eflfective vaccination. 

We have in the earlier illustrations chosen cases in which in all probability a scale 
of character might possibly, if with difficulty, be determined. In the present case, 
the relationship is a very important one, but a quantitative scale is hardly discover- 
able. Nevertheless, it is of great interest to consider what results flow from the 
application of our method. We may consider our two characters as strength to resist 
the ravages of small-pox and as degree of efiective vaccination. No quantitative 
scales are here available ; all the statistics provide are the number of recoveries 
and deaths from small-pox, and the absence or presence of a definite vaccination 
cicatrix. Taking the Metropolitan Asylums Board statistics for the epidemic of 1893, 
we have the table given below, where the cases of " no evidence " have been omitted. 
Proceeding in the usual manner we find 

a^= -86929 a^ = -54157 

h= 1-51139 *= -74145 

c = -782454. 

Hence the equation for r is 

•782,454 = r + '560,310?'^ - -096,3787^ + -081,8817^ - •000,172?'^ - •040,0597'^^ 

whence r = '5954. 

Simiming up we have, after calculating the probable errors, 

h= 1-5114 ± -0287, 
k = -7414 ± -0205, 
r = -5954 db -0272. 

Strength to resist Small-pox when incurred. 



g g 






Cicatrix. 


Becoveries. 


Deaths. 


Total. 


Present 


1562 


42 


1604 


Absent 


383 


94 


477 


Total 


1945 


136 


2081 



O 2 



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44 



PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 



We see accordingly that there is quite a large correlation between recovery and the 
presence of the cicatrix. The two things are about as closely related as a child to its 
" mid-parent." While the correlation is very substantial and indicates the protective 
character of vaccination, even after small-pox is incurred, it is, perhaps, smaller than 
some over- ardent supporters of vaccination would have led us to believe. 

Illustration VIL Effectiveness of Antitoxin Treatment. — ^To measure quanti- 
tatively the effect of antitoxin in diphtheria cases. 

In like manner we may find the correlation between recovery and the administration 
of antitoxin in diphtheria cases. The statistics here are, however, somewhat diflScult 
to obtain in a form suited to our purpose. The treatment by antitoxin began in the 
Metropolitan Asylums Board hospitals in 1895, but the serum was then administered 
only in those cases which gave rise to anxiety. Hence we cannot correlate recovery 
and death with the cases treated or not treated in that year, for those who were likely 
to recover were not dosed. In the year 1896 the majority of the cases were, on the 
contrary, treated with antitoxin, and those not treated were the slight cases of very 
small risk ; hence, again, we are in great difficulties in drawing up a table.* Further, 
if we compare an antitoxin year with a non-antitoxin year, we ought to compare the 
cases treated with antitoxin in the former year with those which would probably have 
been treated with it in the latter year. Lastly, the dosage, nature of cases treated, 
and time of treatment have been modified by the experience gained, so that it seems 
impossible to club a number of years together, and so obtain a satisfactorily wide 
range of statistics. In 1897, practically aU the laryngeal cases were treated with 
antitoxin. Hence the best we can do is to compare the laryngeal cases in two years, 
one before and one after the introduction of antitoxin. The numbers available are 
thus rather few, but will help us to form some idea of the correlation. I take the 
following data from p. 8 of the Metropolitan Asylums Board * Report upon the Use of 
Antitoxic Serum for 1896 ' : — 



Laryngeal cases. 


Eecoveries. 


Deaths. 


Totals. 


With antitoxin, 1896 .... 


319 


143 


462 


Without antitoxin, 1894 . . . 


177 


289 


466 


Totals 


496 


432 


928 



* When a new drug or process is introduced the medical profession are naturally anxious to give every 
patient the possible benefit of it, and patients of coiu*se rush to those who first adopt it. But if the real 
efficiency of the process or drug is to be measured this is very undesirable. No definite data by which to 
measure the effectiveness of the novelty are thus available. 



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Here I find r = '4708 ± '0292. 
A further table is of interest : — 



Laryngeal cases. 


Kequiring 
tracheotomy. 


Not 
requiring it. 


Totals. 


Without antitoxin, 1894 . . . 


261 


205 


466 


With antitoxin, 1896 .... 


188 


274 


462 


Totals 


449 


479 


928 



In this case we have r = "2385 ± -0335. 
Lastly, I have drawn up a third table :— 



Total Infantile Cases, Ages — 5 years. 





Recovery. 


Death. 


Totals. 


With antitoxin, 1896 . . . 


912 


434 


1346 


Without antitoirin, 1894 . . 


615 


556 


1171 


Totals 


1527 


990 


2517 



Here we have* r = '2451 ± 0205. 

The three coeflficients are all sensible as compared with their probable errors, and 
that between the administration of antitoxin and recovery in laryngeal cases is 
substantial. But the relationship is by no means so great as in the case of vaccina- 
tion, and if its magnitude justifies the use of antitoxin, even when balanced against 
other ills which may follow in its train, it does not justify the sweeping statements of 
its eflfectiveness which I have heard made by medical ifriends. It seems until wider 
statistics are forthcoming a case for cautiously feeling the way forward rather than for 
hasty generalisations. 

Illustration VII f. Effect on Produce of Superior Stock. — To find the eflfect of 
superiority of stock on percentage goodness of produce. 

To illustrate this and also the formula (Ixxxiii.) for six correlation coefficients, we wiU 
investigate the eflfect of selecting sire, dam, and one grandsire on the produce when there 

* The values of r for all the three cases of this Illustration were determined with great ease from 
Equation (xxiv.). 



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46 PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 

is selective pairing of dam and sire. We will suppose grandsire, dam, and sire to be 
above the average, and investigate what propoi-tion of the produce will be above the 
average. As numbers very like those actually occurring in the case of dogs, horses, 
and even men, we may take 

Correlation of grandsire and offspring . = '25 

„ sire or dam and offspring = "5 in both cases 

„ sire and grandsire . . . = "5 

Selective mating for sire and dam. . . = "2 

We will suppose zero correlation between paternal grandsire and dam, although 
with selective mating this may actually exist.* We have then the following 
system : — 

^14? ^^ 25, r^ = '5, r^^ = '5, t^ = '2, Vi^ := '5, r^^ = 0. 

Hence, substituting these values in (Ixxxvii.), we find — ^after some arithmetic : 

(Q-Qo)/Qo= 1-4851. 

But Qo is the chance of produce above the average if there were no heredity 
between grandsire, sire, and dam, and no assortative mating. 

N 
Hence it equals iX^X^X^N^— .-. Q= '1553 N. 

Or, of the produce '5 N above the average, '1553 N instead of '0625 N are bom of 
the superior stock owing to inheritance, &c. In other words, out of the '5 N above 
the average, '1553 N are produced by the stock in sire, dam, and grandsire above the 
average, or by '1827 of the total stock, t The remaining '8173 only produce '3447 N, 
or the superior stock produces produce above the average at over twice the rate of the 
inferior stock. Absolutely, the inferior stock being seven times as numerous produces 
about seven-tenths of the superior offspring. 

Illustration IX, Effect of Exceptional Parentage. — Chance of an exceptional 
man being bom of exceptional parents. 

Let us enlarge the example in Illustration II., and seek the proportion of exceptional 
men, defined as one in twenty, born of exceptional parents in a community with 
assortative mating. 

* A correlation, if there be substantial selective mating, may exist between a man and his mother-in- 
law. Its rumoured absence, if established scientifically, would not, however, prove the non-existence of 
selective mating, for A may be correlated with B and C, but these not correlated with each other. 

t The proportion of pairs of parents associated with a grandsire above the average was found by 
putting -5, % and for the three correlation coefficients in (Ixxxv.). In comparing with Illustration II., 
the reader must remember we there deiilt with an exceptional father, 1 in 20, here only vdth relatives 
above the average — a very less stringent selectio^Fi. 



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TO THE THEOEY OF EVOLUTION. 47 

Here we taJce for father and son r^^ = "5, for mother and son i\^ = '5, and for 
assortative mating, r^^ = '2. 

We have then to apply the general formulae (Ixxxiii.) and (Ixxxiv.) for the case of 
three variables. We have 

Ai =^2 =^3 = 1-64485 
A = ^9 = iSa = -484,795 
Ui' = V = V" = 1 '644,850 
Vj' = V," =: t>g"' = 1-705,532 
v,' = <' = V = — -484,356 
v^' = <' = V' = - 5-913,290 

Whence, after some arithmetical reduction, we find 

(Q - QoVQo = 20-0389. 

ButQo = ^X^X-^N = TT^tf^ N. Hence Q = -00263 N. 

We m\ist now distinguish between the absolute and jllative production of excep- 
tional men by exceptional and non-exceptional parents. The exceptional pairs of 
parents are obtained by (xix.), whence we deduce, putting r = -2, ^ = i = 1*64485, 

'^-^ _ A (d + b)(d + c)_ d 1 _ .o^oy, . 

Whence the number of pairs of parents, both exceptional 

= -005245 N. 

Thus, '005245 N pairs of exceptional parents produce '00263 N exceptional sons, 
and '994755 N pairs of parents, non-exceptional in character, produce '04737 N 
exceptional sons, i.e., the remainder of the -^ N. The rates of production are thus as 
'5014 to '0476. Or : Pairs of exceptional parents produce exceptional sons at a rate 
more than ten times as great as pairs of non-exceptional parents. At the same time, 
eighteen times as many exceptional sons are bom to non-exceptional as to exceptional 
parents, for the latter form only about J per cent, of the community. 

The reader who will carefully investigate Illustrations II., VIII., and IX. will grasp 
fully why so many famous men are born of undistinguished parents, but will, at the 
same time, realise the overwhelming advantage of coming of a good stock. 



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[ 49 ] 



II. Electncal Conductivity in Gases Traversed by CatJwde Rays. 

By J. C. McLennan, Demonstrator in Physics^ University of Toronto. 

Communicated by Professor 5. J. Thomson, F.R.S. 

Received December 7, 1899— Read February 1, 1900. 

Though it has been known that a gas becomes a conductor when traversed by cathode 
rays, yet the laws connecting this electrical conductivity have not hitherto been 
studied. 

The theory has been put forward by J. J. Thomson and Rutherford* that when a 
gas becomes a conductor under a radiation, it does so in virtue of the production of 
positive and negative ions throughout its mass. This view has been established by 
their experiments on Rontgenised gases, and confirmed by those of ZELENYf on the 
same subject. The recent work of Rutherford on Uranium Radiation^ also affords 
another example of such a process in the gases traversed. 

The object of the experiments which are described in this paper was to investigate 
the nature of the conductivity in different gases when cathode rays of definite 
strength passed through them, and to measure the number of ions produced. With 
this in view, I have worked with cathode rays produced, after the method of Lenard, 
outside the discharge tube, as these were found to be more easily dealt with than 
those inside. 

The investigation is described under the following subdivisions : — 

1. Form of tube adopted for the production of cathode rays. 

2. Ionization by cathode rays. 

3. Discharging action of cathode rays. 

4. Ionization not due to Rontgen rays. 

5. Discussion of methods for measuring the ionizations produced in different 
(es. 

6. Description of apparatus used. 

7. Explanation of the method adopted for comparing ionizations. 

8. Ionization in different gases at the same pressure. 

9. Ionization in air at different pressures. 

♦ * Phil. Mag.,' November, 1896, p. 393. 
t *Phil. Mag.,' July, 1898, p. 120. 
X *Phil. Mag.,' January, 1899, p. 109. 
VOL. CXCV.— A 263. H 3.11.1900. 



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50 



MR J. c. Mclennan on electrical conductivity in gases 



10. Ionization in a gas independent of its chemical composition. 

11. Comparison of ionizations produced by cathode and by Rontgen mys. 

12. Summary of results. 



1. Form of Tube adopted for the production of Cathode Rays. 

To produce the rays, a modified form of the tube devised by Lenard,* fig. 1, was 
used. The disc a which closed the end and carried the aluminium window formed 
the anode. To hold this disc in position, and to render the joint airtight, recourse 
was had to sealing-wax, which was allowed to set on the previously warmed glass and 
metal, after which the parts were made to unite by slightly melting the surfaces and 
pressing them together. By running round the joint with the pointed flame of a 
blowpipe, any air bubbles present were removed, and complete union was effected. 
Joints made in this way were found to hold for any time desired. 

In making the aluminium window airtight, marine glue could be used, but the 
ordinary commercial soft wax was found to be more suitable. This was especially so 
when the experiments were in the tentative state and alterations were fi'equently 
necessary. The wax melted at a lower temperature than the glue, and besides being 
much more manageable than the latter, it was also less disagreeable to handle. A 
coating of it on the sealing wax also prevented cracking. 

As shown in the figure, the anode was provided with a shoulder round the opening 
of the window. This was found very convenient when the action of the rays on the 




J^i^fi 



air in a partially exhausted receiver such as A was 'oeing examined. The receiver 
was provided with a similar but larger shoulder, and by slipping it over that on the 

* * Wied. Ann.,' vol. 51, p. 225 (1894). 



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TRAVERSED BY CATHODE RAYS. 51 

anode and applying a coating of wax, an airtight connection could be readily made 
without interfering with that which secured the aluminium foil to the disc. This 
latter connection was effected by placing a thin coating of wax upon the brass disc 
and gently applying heat after the foil was laid upon it. All the space within the 
projecting shoulder was then covered with a thick coating of the wax, excepting the 
central portion of the aluminium. 

In all the experiments with these tubes the anode was well earthed, as was also the 
positive terminal of the induction coil used to produce the discharge. 

As regards the distance between the cathode and the anode, it was found best not 
to make it too small. Otherwise, the discharge would pass in the tube before the 
available maximum potential difference was reached. The velocity of the carriers has 
been shown by J. J. Thomson* to vary with the potential difference between the 
electrodes, and as a consequence an intense radiation was more readily obtained when 
the distance between the anode and cathode was considerable. 

In the case of tubes constructed with a short distance between the electrodes, the 
device adopted by McCLELLANDt of inserting an air gap in series with the tube very 
largely increased the intensity of the radiation. 

The foil used by Lenard for the aluminium window was '003 millim. in thickness. 
In practice it was exceedingly difficult to obtain such foil free from holes. Aluminium 
about three times as thick was, however, much better in this regard. The induction 
coil used in the experiments was, besides, very powerful, and, as a radiation 
sufficiently intense could be obtained with it, this thickness was used throughout the 
investigation. 

2. Ionization by CatJwde Rays. 

It has been shown by Lenard| that air, when traversed by cathode rays, acquires 
the property of discharging electrified conductors against which it may be blown, and 
that, fiirther, it retains this property for some time after the rays producing it have 
been cut off. 

According to the theory of Professor Thomson, the air, when in this state, is 
ionized, and the discharging action is brought about by a motion of the ions in the 
gas to the charged conductor. Owing to the separation of the positive and negative 
ions, recombination can take place but gradually, and this readily explains why the 
discharging power is retained by the air for some time. In order to show that these 
positive and negative ions are produced in a gas traversed by the rays, the apparatus 
shown in fig. 1 was used. 

The cathode rays issuing from the aluminium window a passed through a narrow 
tube, 6, into an earth-connected metal chamber, A. B was a disc of brass supported 

♦ *Phil. Mag.,' October, 1897, p. 315. 
t *Proc. Roy. Soc.,' vol. 61, No. 373, p. 227. 
X * Wied. Ann.,' vol. 63, p. 253 (1897). 
H 2 



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52 MR. J. c. McLennan on electrical conductivity in gases 

by an ebonite plug, and surrounded by a guard ring. A wire led from this electrode 
to one pair of quadrants of an electrometer, and the other pair was put to eai-th. 
Care was taken to screen off electrostatic induction by surrounding the wire and 
electrometer with earth-connected conductors. The second electrode, C, also 
supported by an ebonite plug, was connected by a commutator, D, to one of the 
terminals of a battery of small storage cells, the other terminal being connected to 
earth. 

The tube, 6, was made narrow, and penetrated a short distance into the chamber in 
oixler to confine the rays to a slender pencil, and to prevent their impinging upon the 
electrodes. By means of the key, K, the electrode, B, could be put to earth when 
necessary. 

With such an apparatus, and no field initially between the electrodes, it was found 
on exciting the discharge tube and breaking the earth connection, K, that the 
electrometer gained a small negative charge, which did not go on increasing, but soon 
attained a limiting value. 

On the assumption that the cathode rays produce positive and negative ions 
throughout the gas, the explanation of this is obvious. The cathode rays carried a 
negative charge into the gas, and set up a field which caused the negative ions to 
move to the walls of the chamber and to the electrode, B. The charge which the 
latter soon gained, however, set up a field of its own, and a state of equilibrium was 
reached when the conduction to the electrode was just equal to that proceeding from 
it. If, instead of there being no field initially between the electrodes, C was joined 
to the positive terminal of the batteiy, then the electrode, B, gained a positive charge 
when the tube was excited, and the rate at which its potential rose depended upon 
the capacity joined to B and the electrometer. 

With C joined to the negative terminal of the battery, a similar charging took 
place, except that in this case the charge accumulated was a negative one. 

This reversal in the sign of the charge collected may be shown with a field of a 
few volts a centimetre, and clearly points to the existence of positive and negative 
ions in the gas. Since the cathode rays themselves carry a negative charge, the 
presence of these carriers alone in the chamber would account for the negative charge 
obtained with a negative field. With a positive field, however, these carriers would 
be attracted to the electrode C, and it seems impossible to explain how the electrode B, 
under these circumstances, could receive a positive charge unless ions were produced 
by the rays. 

3. Dischargivff Action of Cathode Rays, 

In connection with the experiments of Lenard,* already referred to, cathode rays 
were allowed to fall upon a charged conductor surrounded with air at atmospheric 
pressure. This conductor consisted of a wire attached to a gold-leaf electroscoj^e, 

♦ • Wied. Ann.,' vol. 63, p. 253. 



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TRAVERSED BY CATHODE RAYS. 53 

and was placed within a zinc box in which was a small opening covered with a film of 
aluminium, thin enough to allow the rays to pass through. The end of this wire was 
placed in front of the window and close to it, with the electroscope clear of the direct 
path of the rays. The box itself was connected to earth and set in position, with its 
window opposite that of the discharge tube. 

Using this apparatus, Lenabd found that positive and negative charges alike were 
completely dissipated by a single discharge through the tube when the aluminium 
windows were at any distance up to 4 centims. apart. At greater distances than 
this a similar but only partial discharging of both kinds of electricity occurred when 
the same amount of rays was used. 

This loss of charge was no doubt brought about by means of the ionization in the 
air surrounding the conductor. The known behaviour of an ionized gas, however, 
would have led one to expect a somewhat different result, especially in regard to the 
effect obtained with short distances between the windows. When an insulated metal 
conductor is placed in air ionized by Rontgen rays, Zeleny* has shown that, owing 
to the greater velocity with which the negative ions diflRise, this conductor takes up a 
small negative charge, while the gas itself is left with a positive one. If then the 
ionizations in the two cases are of the same nature, one would have expected that in 
Lenard's experiments the wire and electroscope would not, under any circumstances, 
have been finally discharged completely, but would have been left with at least a small 
negative charge. When, further, it is remembered that the impinging cathode rays 
themselves carried a negative charge to the wue, this fact affords an additional reason 
for expecting such a result. 



Fi^.H. 



Now the gold-leaf electroscope, as used by Lenard (Exner's type), was not 
sensitive to small differences of potential, and it was consequently not a suitable 
instrument for the detection and measurement of effects of this kind. As the 
explanation of his results seemed, then, to be connected with this lack of sensibility 
in the measuring instrument, his experiments were repeated, and a quadrant elec- 
trometer was used in place of the electroscope. 

♦ 'Phil. Mag.,' July, 1S98, p. 13^. 



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54 MR J. c. Mclennan on electrical conductivity in gases 

The arrangement was that shown in fig. 2. A copper wire, terminated by a disc, A, 
of the same metal, was insulated by ebonite fix)m an earth-connected copper tube B, 
through which it passed to the electrometer. To this tube there was fastened, as 
shown in the figure, a large, finely-meshed copper gauze which completely protected 
the disc from electrostatic induction. The tube B also carried a short concentric 
cylinder a, made of copper, which could be slid out when desired so as to surround 
the projecting end of the wire and the disc. 

On placing this apparatus in front of the aluminium window so that the cathode 
rays fell on the disc, it was found that, although the rays caused a discharging of 
positive and of negative electricity, still in no case observed was a negative chai'ge 
on the disc and wire ever completely dissipated. 

Negative charges fell, however, to limiting values, represented in some cases by 
potentials of the order of '25 volt, and then remained stationary. In the case of 
initial positive charges the discharging was not only complete but the disc also gained 
this limiting negative charge. A similar charging action was observed when there 
was no initial charge on the disc. 

Here the disc was subjected to two influences, namely, the cathode rays canying 
a negative charge to it and the ionized gas about it acting as a conductor and tending 
to discharge it. This limiting charge can, then, just as in the case already cited, be 
looked upon as representing a state of equilibrium in which the convection to the 
disc was just equal to the conduction away fix)m it. 

As the electric field produced by a given charge on the disc would vary with 
the distance between it and neighbouring conductors at a different potential, the 
conduction from the wire could consequently be increased or decreased according 
as an earth-connected conductor was brought close to the disc or removed farther 
from it. If then a means were devised of altering in this way the conduction without 
altering the intensity of the rays impinging on the disc, the value of this limiting 
charge could be subjected to definite variations. 

The sliding cylinder a affoixled a simple means of accomplishing this result. If 
when the tube was excited a stationary state was reached, with this cylinder shoved 
well back, and it was then brought forward over the wire and disc, the limiting 
negative charge at once dropped and assumed a steady but smaller value. In order 
to restore the charge to its original value it sufficed merely to slide the cylinder back 
to its former position. 

Another simple verification of this view was afforded by the use of a blast of air. 
If when the rays were impinging on the disc a blast of air was directed towards it 
and at right angles to the rays, the limiting charge at once increased to another 
limiting valut^, and when the blast stopped it again dropped to its original amount. 

As the velocity of the cathode rays has been estimated by J. J. Thomson* to be of 
the order of 10^^ centims. per second, it is clear that any ordinary bliist could produce 

♦ * Phil. Mag.,* October, 1897, p. 315. 



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TRAVERSED BY CATHODE RAYS. 55 

very little effect on the motion of these carriers. On the other hand, the velocity of 
the ions in Rontgenised air has been found by Rutherford* to be about 1 '6 centims. 
per second, under a field of a volt a centimetre, and consequently of the order of that 
of the blast. In the experiment described, the effect of the blast, therefore, was to 
decrease the conduction away from the electrode by removing the ionized gas ; and as 
no change was made in the intensity of the rays impinging on the disc, this con- 
sequently produced an increase in the residual charge. This increase, however, did 
not go on indefinitely, but ceased when the field it set up was sufficient to neutralise 
the effect of the blast ; hence the second stationary value for the charge. 



Another means of increasing this limiting charge was afforded by the removal of 
the air surrounding the electrode. To show this the gauze cap was removed from 
the apparatus in fig. 2, and the metal tube siurounding the wire was brought forward 
and sealed to the anode of the discharge tube. The arrangement is shown in fig. 3, 

With this apparatus it was found that, as the exhaustion proceeded in the chamber 
B, the negative charge received by the electrode A gradually increased, until finally, 
at a very high vacuum, a momentary discharge of the rays was sufficient to raise its 
potential beyond the range of the electrometer. This result, therefore, confirms the 
explanation already given of the discharging action of the rays. In a recent paper 
by LENARDt this charging action of the cathode rays in a high vacuum was described, 
but its connection with the ionized air surrounding the electrode was not brought 
out. From the experiments just described it is clear that, while this action is directly 
due to the fact that the cathode rays carry a negative charge, the extent of the effect 
obtained in all cases depends to a very great degree upon the opposing influence 
exerted by the ionized air surrounding the electrode upon which the rays fall. 

4. Ionization not due to Rontgen Rays. 

It has been thought by some that the ionization pixxluced by c«.thode rays was due 
to Rontgen rays, which might possibly be sent out from the window at the same 
time. The results of experiment are, however, entirely opposed to this view. 
* *Phil. Mftg.,' Noveml>er, 1897, p. 436. t * Wied. Ann.,' vol. 63, p. 253. 



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56 MR. J. c. Mclennan on electrical conductivity in gases 

In order to investigate the point an apparatus similar to that shown in fig. 3 
was adopted. Different thicknesses of aluminium foil were in turn used for the 
window, and the air in the chamber B was kept at a pressure low enough to 
absorb but little of any radiation coming from the window, and yet sufficiently 
high to aflford considerable conductivity when ionized. 

With diflferent thicknesses of the foil down to "04 millim., it was found that the 
electrode A did not gain any charge when the tube was excited. Further, if in 
these cases a charge, either positive or negative, was given independently to the 
electrode, this charge was maintained when the discharge passed in the tube, and 
no leak occurred. But when the window was made of foil '008 millim. in thickness, 
the effect obtained was such as that already described in the last paragraph. 
Under these conditions the electrode A, if carrying initially a positive or a negative 
charge, finally assumed a stationary state, in which it carried a definite negative 
charge whose value, as has already been pointed out, depended upon the pressure 
of the air in the chamber B. As, then, no leak from the electrode occurred when 
the aluminium was '04 millim. in thickness, it seems justifiable to conclude that if any 
Rontgen rays were present under these circumstances they were of an extremely 
weak character. If Rontgen rays of even very moderate intensity had entered the 
chamber, a leak would have taken place which could have been observed. In practice 
the aluminium foil used in my experiments was about '008 millim. in thickness, and 
with this foil intense ionization was observed. From the known character of Rontgen 
rays, it was quite impossible for this great ionization to be produced by rays which 
could be absorbed by a layer of aluminium "032 millim. — ^the difference in thickness 
of the two windows. 

Again, an ordinary focus tube illustrates very well the fact that the Rontgen 
rays produced issue in a large measure firom the face of the anticathode, upon 
which the cathode rays fall, while the radiation appearing to come fi:'om the opposite 
face is always very weak. The theory now generally accepted is that the Rontgen 
rays are electromagnetic pulses sent through the ether when the moving electrified 
particles which constitute the cathode rays are suddenly stopped. If then the 
Rontgen radiation sent out in the direction of propagation of the cathode rays, when 
these carriers were stopped by foil '04 millim. in thickness, was at most but very 
feeble, it appears highly improbable that a strong radiation of this kind could be 
produced by those carriers that passed through the thinner foil without being 
stopped. 

The conductivity produced in a gas by cathode rays is, moreover, far in excess of 
that excited by even the strongest Rontgen rays. In order to make a direct com- 
parison, measurements were taken of the ionizations produced in the same chamber 
by both radiations, and the following illustration gives an indication of their 
respective efficiencies. By using the apparatus shown in fig. 1, it was found that, 
under the action of cathode rays with a saturating intensity of field, a capacity of 



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THAVEBSED BY CATHODE RAYS. 57 

750 electrostatic units attached to the electrode B gained in 15 seconds a charge 
represented by 300 divisions on an arbitrary scale. A Rontgen ray focus tube 
giving out very strong rays was then used in place of that for producing the 
cathode rays, and was excited by an induction coil capable of giving a 50-centims. 
spark. Under these circumstances, with the same field, which was also in this case 
a saturating one, a capacity of 150 electrostatic units was charged in one minute to 
an amount represented by 20 on the same scale. This case, which is an extreme 
one, shows that the ionization by cathode rays was about 300 times that due to an 
intense Rontgen radiation. In the present investigation these latter rays, even if 
they did accompany the cathode rays, must have been very feeble, and could there- 
fore only exert an ionizing influence which may be left out of consideration. 

The known action of a magnetic field naturally suggested itself as a means of 
sifting out the cathode from any accompanying Rontgen rays. The intensity of the 
cathode rays, however, soon falls off owing to their rapid absorption by the air, and 
on this account it was necessary to place the chamber in which the ionization was 
measured close up to the discharge tube. Under these conditions it was found 
impossible to deflect the rays outside the tube without also deflecting those inside. 
This difficulty consequently rendered the test indecisive, and the method had to be 
abandoned. 

5. Discussion of Methods for Measuring the Ionizations produced in Different Gases. 

In the construction of Rontgen-ray bulbs, the disengagement of gas from the 
electrodes and the inside of the glass is facilitated by the application of heat to the 
tube. In the case of Lenard tubes, however, the joints are made of wax, and the 
final stage of exhaustion cannot be hastened by adopting this device. In practice a 
tube was kept attached to the mercury pump, and exhausted while the discharge 
was passing through it. After some hours of this procedure the coil was stopped, 
and the exhaustion was continued until only some traces of air were being taken 
over. On then exciting the tube, the vacuum was found to be sufficiently high for 
the cathode rays produced to penetrate the aluminium window. After running the 
coil for a short time, a small quantity of gas acciunulated in the tube, and the 
pressure rose so high that the rays ceased to be propagated outside. After this air 
had been removed the vacuum again became good, and the original intensity of tlie 
rays was restored. As the ionizing power of the rays was very great, charges 
sufficiently large to be accurately measured were easily accumulated by exciting the 
tube only for short periods. By following this course quite satisfactory results 
were obtained and much loss of time was avoided. 

On account of this running down of the discharge tube, it was impossible, in 
comparing the ionizations in two different gases, to use an apparatus with a single 
chamber, such as that shown in fig. 1. In order to obtain accurate results, it was 

VOL. CXCV. — A. I 



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58 MR. J. c. McLennan on electrical conductivity in gases 

necessary either to have a constant source of rays, or else to be able to ascertain the 
relative intensities of the rays used with the different gases. 

One method which suggested itself was the use in series of two chambers, such as 
that shown in fig. 1. By inserting a thin aluminium membrane between them a 
different gas could be put in each chamber, and a single pencil of rays could be used 
to produce the ionization in both chambers. With this arrangement it was thought 
that the ionization obtained in the first chamber might perhaps bear a constant 
ratio to that produced in the second. But this relation was not found to hold, and 
further, as the cathode rays are rapidly absorbed, the amount of ionization obtained 
in the first chamber was so very much greater than that in the second, that even 
if the ratio had been fairly constant the method would not have been at all 
satisfactory. 

This led to a trial of two receivers in parallel. Although the cathode rays on 
issuing from the window diverge very greatly, mechanical difficulties made it im- 
practicable to receive part of the issuing rays in each chamber, and so recourse was 
had to the use of two windows. With a single large disc as cathode, a stream of 
rays was received in each of the chambers. The ratio of their intensities, however, 
as measured by the ionizations they produced, did not remain constant but varied 
quite irregularly. The explanation of this is probably found in a paper by A. A. C. 
SwiNTON,* where he points out that the carriers are shot off in a hollow cone from 
the cathode, and that the dimensions of such a cone of rays vary with the degree oi 
exhaustion in the tube. Besides, the aluminium windows were opposite to eccentric 
points on the cathode, and the ratio of the intensities of the two pencils was in this 
way greatly influenced by slight variations in the directions of the rays within the 
tube. A cathode formed of two small discs was then tried, and the results obtained 
were very satisfactory. The ratio of the discharges from the windows was in this 
case quite constant, and it was therefore possible to make measurements with con- 
fidence. The main difficulty of the investigation was in this way overcome, and the 
method was applied to obtain among other things a knowledge of — 1, the absorption 
of the rays ; 2, the ionizations produced by them in air at different pressures ; and 
3, the relative ionizations in different gases. 



6. Description of Apparatus used. 

A diagram of the apparatus is shown in fig. 4, and the way in which the connec- 
tions were made is exhibited in fig. 5. The exciting tube was slightly over 
3 centims. in diameter. The two discs of the cathode were each about a centimetre 
in diameter, and they were placed with their centres directly in front of the 
alimiinium windows. That portion of the apparatus in which the ionizations were 

* *Proc. Roy. Soc.,' vol. 61, p. 79 (1897). 



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TRAVERSED BY CATHODE RAYS. 59 

measured consisted of two chambers, A and B, each made of brass and similar in 
form to that shown in fig. 1. The two electrodes C and D were held in position by 
ebonite plugs, which closed the ends of the receivers and at the same time served as 
insulators. 




el*^ 



$arf^ 



In each experiment the receivers themselves were well earthed, and also, initially, 
the electrodes C and D. As the electrostatic induction was very intense in the 
neighbourhood of the discharge tube, it was found necessary to take special 
precautions in regard to the earth connections. Wires of but very small resist- 
ance were used, and these were led to water mains and all the joints carefully 
soldered. 

The two chambers were separated by a disc of ebonite, and to its faces were 
attached thin brass plates, a and h. By means of wires passing out through the 

I 2 



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60 



MR. J. c. Mclennan on electrical conductiyity in gases 



ebonite to the battery of storage cells E, these plates could be charged to any 
desired potential, high or low. As the electrodes and the walls of the receiver were 
earthed, this afforded a means of setting up in each chamber a field which could be 
readily modified. The fields themselves, moreover, were quite distinct, each disc 
serving as a screen to cut off any action arising fi'om the other. 

Each of the chambers was provided with a projecting shoulder, which slid over a 
corresponding one on the anode surrounding the window opposite. By coating these 
joints with wax the chambers were then not only made airtight, but also were 
entirely separated from each other. 

In the apparatus used, the diameter of the chambers A and B was about 3 centims., 
and the distance between each of the electrodes and its corresponding plate a or 
h about 1'6 centims. The diameter of the narrow cylinders which admitted the rays 
to the chambers was 3 millims., and the distance between the aluminium windows 
and points corresponding to the centres of the electric fields was about 2 centims. 



\>XNXX\XXX\XX\\\\\\\\\\\\N>X\\X\ 




Each of the electrodes was connected to an air condenser, whose capacity was 
about 600 electrostatic units. These condensers, G and H, were each made of two 
sets of parallel plates separated by small ebonite supports. The plates were made by 
coating both .sides of a sheet of glass by a single sheet of tinfoil. In this way plates 
tolerably plane were obtained, and yet difficulties arising from electric absorption 
were avoided, the glass merely serving as a support for the foil plates. 

The measurements were made with a quadrant electrometer F, and the tube was 
excited by a 50-centims. spark-length induction coil, whose positive terminal, together 
with the anode of the tube, was kept to earth. This coil was provided with an 
Apps interruptor, and besides being very powerful was also very efficient. It 



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TRAVEESED BY CATHODE RAYS. 61 

required a potential of only eight volts to excite it, and with the interruptor working 
slowly, this was suflScient to produce sparks of the maximum length in air at normal 
pressure. In practice, the interruptions were made at the rate of 20 to 25 per 
second. 

7. Explanation of the Method adopted foi* Comparing Ionizations. 

It is well known that, in conduction in Rontgenised gases, and in gases acted upon 
by uranium radiation, the current of electricity obtained does not increase in propor- 
tion to the electromotive force applied. The current, after reaching a certain critical 
value, becomes practically stationary, and increases but very little when very large 
increases are made in the electromotive forces. This maximiun, or saturation current, 
was also found to characterise the conductivity produced by the passage of cathode 
rays through a gas. With Rontgen or uranium radiation, a field of 400 or 500 volts 
a centimetre has been found to give saturation in most simple gases; but with 
cathode rays it was necessary to apply fields of much stronger intensity. 

As already stated, the distance between either of the electrodes C and D, fig. 5, 
and the dividing partition was about 1 '6 centims. In order to ascertain the saturating 
electromotive force, the plate h was kept at a very high potential, while that of 
a was gradually increased fi'om zero. At each stage the ratio of the currents 
obtained in the two chambers was noted, and it was not until a potential of about 
900 volts was applied to a that an approximation to the saturation current was 
obtained in the chamber A. With a potential difference of 1200 volts the increase 
in the current was small, and an increase only slightly larger was obtained with a 
potential of 1600 volts, or 1000 volts a centimetre. This small increment in the 
current very probably arose from the influence of the field itself. It may be that in 
certain parts of the receiver the rays, acting in conjunction with the applied differ- 
ence of potential, had not quite sufl&cient intensity to produce dissociation. An 
increase in the field under these circumstances would produce greater ionization, and 
consequently a larger current would be obtained. As this field of 1000 volts a centi- 
metre practically produced saturation currents in both chambers, it was used through- 
out in measuring the ionizations. Sparking was prevented by using in the charging 
circuit liquid resistances, such as xylol. 

An explanation of the saturation current is that the number of ions used up by the 
current in a given time is exactly equal to the number .produced by the rays in the 
same time, or in other words, the ions are removed so rapidly by the applied field 
that recombination is practically eliminated. The saturation current is then a direct 
measure of the ionization produced, and in order to compare the ionizations in any 
two gases, it suffices to measure their saturation currents. In this investigation the 
saturating electromotive force was applied to the plates a and 6, the discharge 
tube was then excited, and the currents obtained were used to charge up the con- 



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62 MR J. C. McLENNAN ON ELECTRICAL CONDUCTIVITY IN GASES 

densers G and H. The discharge having been stopped, the potentials of the two 
condensers were then successively determined. 

As the effective capacity of the electrometer was the same fraction of that of each 
of the two equal condensers, the deflection readings were direct measures of the 
charges obtained. The charging of both condensers proceeded for the same time, 
and consequently the electrometer deflections were also direct measures of the 
saturation currents, and therefore of the ionizations in the two chambers. 

The method possessed the advantage of being independent of the time of charging 
and of the strengths of the rays coming from the two windows, provided only that 
the ratio of their intensities remained constant. In using the electrometer the 
needle was kept at a high potential, and one pair of quadrants always connected to 
earth. Though with this arrangement slow losses from the needle occurred, yet the 
short interval required for the two readings made the gradual change in the effective 
capacity of the electrometer inappreciable. 

In practice, the electrometer was initially connected to one of the condensers, and 
the tube allowed to run until a suitable deflection was obtained. After noting this 
reading, the electrometer, having been put to earth, was then connected to the other 
condenser and the second reading taken. In this way the ratio of the ionizations in 
the two chambers was obtained. 

From the experiments described in Section 2, it is clear that the signs of the 
charges obtained in the condensers depended on the signs of the charges given to 
a and h by the battery. In case these plates were positively charged, the 
charges collected were positive, and were due entirely to ionization. With a nega- 
tive field, however, the negative charges obtained included not only negative ions 
produced by the rays, but also the negative carriers, constituting the rays, that were 
stopped in their motion by the gas. For this reason the positive field was always 
used, and consequently the charges obtained gave a measure of the number of ions 
produced in the gas by the passage of the rays. 

8. Ionization in different Gases at the Same Pressure. 

To compare the ionization in a selected gas with that in air at the same pressure, 
the saturating electromotive force was applied to the plates a and &, fig. 5. The 
two chambers A and B were first filled with air at atmospheric pressure, and a series 
of readings taken, the mean of which gave the ratio of the saturation currents in 
the two chambers. The air was then removed from A, and the gas to be tested 
introduced. A set of readings similarly taken gave a ratio for the saturation 
current obtained with the given gas in A, compared with that obtained with air in 
B. The combination of these results gave the ratio of the saturation current in A, 
when filled with the given gas, to that in the same chamber when filled with air. 
This ratio was, consequently, the ratio of the ionization produced in the selected gas 



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to that produced in air at an equal pressure under the action of cathode rays 
entering the chamber with the same intensity in both cases. 

The results obtained from this method for hydrogen, air, and carbon dioxide are 
given in the first column of Table I. In the second column are given the relative 
ionizations found by J. J. Thomson* for these same gases when ionized by Rontgen 
rays of constant intensity. 

Table I. 



Name of gas. 


Column I. 


Column 11. 


Ionization by 
cathode rays. 


Ionization by 
Rontgen rays. 


Hydrogen 

Air 

Carbon dioxide . . . 


2-65 

1-00 

•34 


•33 

100 
1-40 



These numbers, it will be seen, present a very marked difference. In the one case 
the ionization decreased as the density of the gas traversed increased, while in the 
other a law directly the reverse of this was followed. 

One explanation of this difference in the results is that the character of the 
ionization under cathode rays may be essentially different from that produced by 
Ilontgen rays. Apart from these numbers, however, there seems to be but little 
ground for this view. Strong experimental evidence now exists to support the 
assumption that the cathode rays consist of small particles of matter carrying nega- 
tive charges of electricity. We may therefore regard the ionization they produce as 
being due to their impinging on the molecules of a gas, and to the consequent 
breaking up of the latter. On this hypothesis it is not clear that the resulting ions 
should differ in character from those produced under the influence of Rontgen 
radiation. 

It appeared rather that the true explanation was to be found in the varying 
absorbing powers of the different gases. LENARD,t who studied these rays by the 
fluorescence they excited, found that the absorption of cathode rays by gases at 
atmospheric pressure was considerable. He was also led by his experiments to 
propound the law, that while different gases at the same pressure absorbed the rays 
to different degrees, yet their absorption depended only upon the densities of the 
gases, and not upon their chemical composition. 

In the apparatus here used, the distance traversed by the rays after they left the 
discharge tube until they reached the centre of the field where the ionization was 

♦ *Proc. Camb. Phil. Soc.,* vol. 10, Part I., p. 12, 
t • Wied. Ann.,* vol. 56, p. 265 (1895). 



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64 



MR. J. c. Mclennan on electrical conductivity in gases 



measured, was about 2 centims. From Lenard's conclusions, it is obvious that in this 
distance the absorption of the rays by carbon dioxide would be greater than by air, 
and very much greater than by hydrogen. The effective intensities of the rays in the 
three gases at the same pressure would then be very diflferent, and numbers such as 
those given in Column I. follow naturally under these circumstances, without assuming 
any difierence in the character of the two ionizations. 

9. Ionization in Air at Different Pressures. 

In order to study more closely the influence of absorption, a number of experiments 
were carried out similar to that just described. The same apparatus was used, and the 
same method followed, but the ionizations, instead of being measured in different 
gases at the same pressure, were determined for the same gas at different pressures. 



Table IL 





Pressure. 


Ionization measured. 




millims. 






767 


1-00 




530 


1-44 




340 


1-92 




205 


2-32 




104 


2-68 




53 


2-74 



Between 40 and 45 millims. a sudden large 
increase was obtained in the ionization. 
This was found to be due to the action 
of the field itself in dissociating the gas. 



The results obtained with air are shown in Table II. The pressures are expressed 
in heights of columns of mercury at the same temperature. The ionizations given 
are relative, that corresponding to atmospheric pressure being taken as unity, and 
each value is the average of a large number of readings. 

The results are also shown graphically in fig. 6, where the abscissae represent 
pressures, and the ordinates corresponding relative ionizations. 

The numbers show that as the pressure decreased the ionization obtained with a 
saturating electromotive force steadily increased, until a pressure of about 75 millims. 
of mercury was reached. This result, though surprising, can be readily explained 
by the great absorption of the rays at atmospheric pressure. 

The rays had to travel at least 1 "5 centims* from the window before they reached 
that part of the chamber from which the saturation current was obtained. For this 



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TRAVEliSED BY CATHODE RAYS. 



65 



reason their efTective intensity was very largely determined by the pressure of the 
gas traversed. 

While a diminution in the pressure would not affect the original intensity of a 
pencil of rays issuing from the window, it would, owing to a decrease in the absorp- 
tion, increase the ionizing power of this pencil at the centre of the receiver. In this 
way, although the available amount of matter to be ionized was lessened by lowering 
the pressure, it could happen that the resultant ionization, as measured by the satu- 
ration current, would at first exhibit increasing values. This in all probability 
accounts for the numbers obtained in Table H. 

Now frora this point of view such a condition would only hold down to a stage 
when the two influences produced equal effects. The ionization would then be a 
maximum, and would afterwards fall off with diminishing pressures. Although the 
numbers obtained for the saturation current do not show definitely that a maximum 
value was obtained for the ionization, still there are indications from them, as the 
curve shown on fig. 6 illustrates, that the maximum value was reached at a pressure 
of about 75 millims. of mercury. 



/on/zat/ofTS 



4 


































































3 


























^^ 


.-i^ 






2 

/ 


^.^ 


^ 


















\ 


lOO ?( 


90 


fi 


00 


5 


10 


m 


H> 


— 


70 


"T 


55" 


A 


W i 


} 





PRsaauRcs IN MMS. 



Fig. VI. 



As indicated in Table II., the conditions of the experiment made it impossible 
to measure the ionization in air at pressures much below 50 millims. At about 
40 millims. pressure a sudden large increase was obtained in the value of the satura- 
tion currrent, whch was found to be due to the influence exerted by the applied field 
in breaking down the gas. At these low pressures the electric intensity, which was 
1000 volts a centimetre, wassufiicient to dissociate the attenuated gas and to produce 
a discharge on its own account between the electrodes. This was shown by simply 
connecting the electrometer to one of the electrodes, C for example, and applying the 
potential difference without exciting the discharge tube. On then exhausting the 



VOL. CXCV. — A. 



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66 MR. J. c. McLennan on electrical conductivity in gases 

chamber, the electrometer showed no leak until the critical pressure was reached, 
when it inmiediately began to charge up. 

10. Ionization in a Gas independent of its Chemical Composition. 

An important result in connection with these experiments is the agreement 
exhibited between the number given in Table I. for the ionization in hydrogen at 
atmospheric pressure and that given in Table II. for the ionization in air at a 
pressure of 53 miUims. 

Here two gases, hydrogen and air, were introduced in succession into the same 
measuring chamber and adjusted to the same density. Cathode rays of the same 
intensity were projected into this chamber in the two cases, and these rays, after 
traversing a certain length of the gas, reached a point where the ionization they 
there produced was measured. The values obtained show that under the circum- 
stances the same number of ions was produced in both gases. 

Since the rays issuing from the window were in both cases of the same intensity, it 
follows from Lenard's absorption law that the disposition of the rays, their actual 
intensities, and the quantities of them absorbed from point to point in the chamber, 
were precisely the same in both gases. Under these circumstances, therefore, the 
equal ionizations obtained in hydrogen and in air at the same density not only form 
a confirmation of Lenard's absorption law, but also show that where equal absorption 
occurs equal ionization is produced. 

In the case of Rontgen radiation, Rutherford* has made a determination of the 
relative absorbing powers of a number of gases. Taking I to denote the intensity 
of the rays on entering a particular gas, and le"^ their intensity after traversing a 
length X, he has found that the values of the coefficient of absorption for the different 
gases practically represented the relative conductivities produced in these same gases 
by Rontgen rays. It is thus interesting to note that with cathode rays, just as with 
Rontgen rays, equal absorption gives equal ionization. 

To test still further the accuracy of this conclusion a detailed examination was 
made of the ionization produced in a number of different gases. Throughout the 
experiments air in the chamber B, fig. 5, was taken as the standard. In some 
comparisons this air was kept at atmospheric pressure, while in others lower 
pressures were taken, the pressure selected being maintained through each complete 
determination. In making a comparison the chamber A was fiJled in turn with the two 
gases to be examined, and their pressures were adjusted so as to reduce them to the 
same density. Two ratios were in this way found for the ionizations in the chambers 
A and B, and as the influence of absorption was eliminated on account of the equal 
densities, these ratios represented the relative ionizations in the two gases under 
cathode rays of the same intensity. 

* * Phil. Mag.,' April, 1897, p. 254. 



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67 



These ratios were determined by taking the mean of a number of readings. 
Samples of the results obtained in five different comparisons are given in Tables III., 
IV., v., VL, and VIL, the numbers being consecutive readings with each gas in A. 
They represent very well the working of the method. Although the variations were 
considerable, similar ones occurred in both sets of observations in each comparison, and 
as the number of readings taken was very large, any errors were in a great measure 
compensated. 

Table III. — Oxygen and Air. 



Air in both chambers at 


Oxygen in A at 675-1 millims. 


746-7 millims. 


Air in B at 746-7 millims. 


Ionization in A. 


Ionization in B. 


Ionization in A. 


Ionization in B. 


109 


100 


1-40 


1-00 


1-37 




1-28 


99 


1-54 




1-39 




9 


1-24 




1-20 




» 


1-35 




1-54 




1 


1-25 




1-41 




} 


1-07 




1-20 




9 


1-41 




1-26 




9 


1-54 




1-41 




9 


1-31 




1-33 




9 


1-32 


100 


1-34 


1-00 



Table IV. — Nitrogen and Air. 



Air in A at 734-3 millims. 


Nitrogen in A at 757 millims. 


Air in B at 757 millims. 


Air in B at 757 millims. 


Ionization in A. 


Ionization in B. 


Ionization in A. 


Ionization in B. 


1-10 


1-00 


1-04 


1-00 


1-02 


99 


M2 




1-21 


J 


J 


1-03 




1-05 


J 


9 


1-34 




1-29 




i 


1-03 




110 




9 


106 




1-18 




9 


M5 




1-04 




9 


M2 




1-07 




9 


1-08 




1-00 




n 


1-06 




Ml 


1-00 


MO 


1-00 



K 2 



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68 



MR. J. C. McLENNAN ON ELECTRICAL CONDUCTIVITY IN GASES 



Table V. — Carbon Dioxide and Air. 



Air in both chambers at 
772-7 millims. 


Carbon dioxide in A at 

504-7 millims. 
Air in B at 772-7 millims. 


Ionization in A. 


Ionization in B. 


Ionization in A. 


Ionization in B. 


1-22 


100 


1-17 


1-00 


1-12 


)) 


116 


» 


1-17 




) 


1-23 


9) 


1-3S 




} 


1-31 


)) 


1-02 




» 


1-37 


}) 


1-30 




1 


1-00 


yj 


111 




y 


1-21 


)) 


117 




y 


1-24 


)) 


103 




f 


1-31 


yy 


1-23 




> 


100 


i» 


1-17 


1- 


00 


1-20 


1-00 



Table VI.— Hydrogen and Air. 



Air in both chambers at 


Hydrogen in A at 770*9 millims. 


53*2 millims. 


Air in B at 53*2. 


Ionization in A. 


Ionization in B. 


Ionization in A. 


Ionization in B. 


1-58 


1-00 


1-52 


100 


1-77 




1-82 


}} 


1-64 




1-91 




) 


1-41 




1-63 




y 


1-62 




1-58 




y 


1-63 




1-80 




y 


1-79 




1-70 




y 


1-73 




1-32 




y 


1-85 




1-75 




y 


1-81 




2-04 




> 


1-68 


1-00 


1-71 


100 



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69 



Table VII. — Nitrous Oxide and Air. 



Air in both chambers at 
759 minima. 


Nitrous oxide in A at 

499 millims. 

Air in B at 759 millims. 


Ionization in A. 


Ionization in B. 


Ionization in A. 


Ionization in B. 


1-08 


1-00 


103 


100 


MO 






1-21 


99 


1-24 






M5 


9) 


112 






1-07 


)) 


•99 






113 


11 


1-08 






1-07 „ 1 


111 






117 


99 


1-23 






1-02 


99 


1-07 






1-05 


)) 


112 






110 


»> 


111 


1-00 


1-10 


100 



Care was taken to insure the purity of the gases, and they were also well dried 
before being passed into the ionizing chamber. 

The oxygen was prepared electroly tically, and was freed from ozone by being passed 
through a strong solution of potassium iodide and caustic potash. 

The nitrogen was prepared by gently heating a mixture of ammonium chloride with 
a nearly saturated solution of sodium nitrite. The gas given off was passed through 
a U-tube containing strong caustic potash, and also through a second containing 
concentrated sulphuric acid. A Kipp apparatus was used for the preparation of carbon 
dioxide, which was made in the ordinary manner by allowing dilute hydrochloric acid 
to act on marble. In making hydrogen a Kipp apparatus was also used, dilute 
sulphuric acid being allowed to act on zinc. The gas was passed through a strong 
potassium permanganate solution, and then through a U-tube containing a strong 
solution of caustic potash. 

The nitrous oxide was prepared by heating ammonium nitrate in a flask, and the 
gas was collected over water. 



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70 



MR. J. c. McLennan on electrical conductivity in gases 



Table VIII. — Summary of Measurements. 



Gases compared. 


Pressures. Ionizations. 

1 


Air mean of 30 readings 

Oxygen „ 30 „ 


millim*. 

746-7 

675-1 


1-31 
1-32 


Air mean of 25 readings 

Nitrogen .... » 25 „ 


734-3 
757 


Ml 
1-09 


Air mean of 30 readings 

Carbon dioxide . . « 30 „ 


772-7 
505-4 


1-20 
118 


Air mean of 18 readings 

Hydrogen .... „ 18 „ 


53-2 
770-9 


1-70 
1-79 


Air mean of 23 readings 

Nitrous oxide . . i, 23 „ 


759 
499-3 


1-09 
1-10 



A summary of complete sets of observations on the different gsises is given in 
Table VIII. This statement includes the number of readings made in each case and 
the pressures at which these were taken. The ionizations quoted are the averages of 
the several sets of readings. 

The close agreement exhibited by the numbers corresponding to each comparison 
fully bears out the conclusion deduced from the earlier experiments. It not only 
forms a striking corroboration of Lenard's absorption law, but also shows that the 
ionization follows an analogous one, which may be stated thus : — When cathode rays 
of a given strength pass through a gas, the number of ions produced per second in 
1 cub. centim. depends only upon the density of the gas, and is independent of its 
chemical composition. 

The similarity in the laws of absorption and ionization, holding, as it does, with so 
many gases over such a wide range of pressures, is a clear indication that when 
cathode rays are absorbed to a certain extent, the positive and negative ions produced 
by these absorbed rays are of a definite amount, which bears a constant ratio to the 
quantity of the rays absorbed ; that is to say, the absorption of a definite amount of 
radiant energy is always accompanied by the appearance of a fixed amount of potential 
energy in the form of free ions. 

This granted, it follows that in order to ascertain the relative ionizations pro- 
duced in any two gases by cathode rays of the same intensity, it is sufficient to 
determine the absorbing powers of the two gases for the same rays. In other words, 



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TRAVERSED BY CATHODE RAYS. 71 

the coefficients of ionization for a series of gases are fully determined when the 
coefficients of absorption for these same gases are known. 

The existence of this general relation between absorption and ionization for both 
cathode and Rontgen rays is especially interesting when we remember that the two 
radiations are so very different in many respects. 

In the one case, according to the generally-accepted view, the rays consist of 
small charged particles of matter moving with high velocities in space, while in the 
other they are supposed to consist of electromagnetic impulses propagated in the 
ether. With the one the dissociation is in all probability brought about by a series of 
impacts between the moving particles and the molecules of the gas ; with the other 
it seems to be due to the direct action of the intense electric field forming the impulse. 
Again, while the absorption of cathode rays depends only upon the density of the 
medium traversed, the absorption of Rontgen rays, according to Rutherford's 
results, does not seem to depend to any great extent upon the molecular weight of 
the gas. But while all these differences exist in the two radiations, with both of 
them it holds good that the same number of ions are always produced in a gas when 
the same amount of rays traversing it are absorbed. 

11. Comparison of Ionizations produced hy Cathode and hy Rontgen Rays. 

The method just described gives definite and conclusive information regarding the 
ionizations produced by cathode rays in gases of the same density ; but where the 
gases are of different densities, it cannot be satisfactorily applied. As stated in 
Section IX., the rays, after entering the ionizing chamber, must travel some distance 
before reaching that part of the field fi'om which the current is drawn. On this 
account, though rays entering the chamber may originally be of the same strength, 
still their effective intensities become at ordinary pressures quite different, when the 
gases traversed are not of the same density. 

Also as it is impossible to define exactly the disposition of the electric field within 
the chamber, these effective intensities cannot be calculated with any degree of 
accuracy. 

A difficulty arises, too, from the dispersion of the rays. As shown by Lexard, 
they issue from the window in a pencil whose form is greatly influenced by the 
density of the gas traversed. At very low pressures they pass through the 
aluminium window practically without deviation, but as the pressure increases, they 
spread out until finally they issue in all directions. 

The conclusion arrived at in the last section, however, suggests a means of calcu- 
lating the ionization which would be produced by rays of constant intensity in 
different gases at the same pressure. 

Lenard,* who investigated the absorption powers of a number of gases at different 

* * Wied. Ann.,' vol. 56, p. 258. 



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72 



MR. J. c. mclen:nan on electrical conductivity in gases 



pressures, has shown that for any particular gas the coefficient of absorption varies 
directly as the pressure. In the case of air, taking I to denote the intensity of the 
rays issuing from the window of the discharge tube, and le"^ their intensity at a 
distance x from the window, he found for \ the values given in Table IX. 





Table IX. 


Air pressure 


• 


Coefficient of absorption. 


millims. 






760 




3-43 


331 




1-51 


165 




•661 


83-7 




•396 


40-5 




•235 


19-3 




•117 


100 




•0400 


2-7 




•0166 


•78 




•00416 



These numbers, it wiU be seen, amply support Lenard's conclusion. Similar 
tables, given by him for a number of gases, all exhibit the same relation between the 
values of X and the corresponding pressures of the gas. 

Now, if the values of the coefficient of absorption are taken to represent the rela- 
tive ionizations produced in a gas, at a point where the pressure is varied but the 
intensity of the rays kept constant, it follows from Lenard's numbers that the 
ionization in any particular gas would vary directly as the pressure to which it was 
subjected. 

This result, which follows as a deduction from the preceding experiments, has also 
been found experimentally by Perrin* to characterise the ionization produced by 
Rontgen rays. It is true that with Rontgen rays a number of experimenters have 
found quite different relations to hold between the ionization and the pressure ; but 
in most cases they have vitiated their results either through omitting to use satu- 
rating electromotive forces, or through neglecting to arrange their experiments so as 
to eliminate the metal effect observed by Perrin. 

With uranium radiation also, RuTHERFORDt has found the ionization to be propor- 
tional to the pressure of the gas traversed. 

The direct experimental verification of a law of this kind is always accompanied by 
a serious difficulty. The law has reference to the action of rays whose intensity is 
constant throughout the region ionized. With rays that are easily absorbed by 
gases at ordinary pressures, this condition can be realised either by the use of very 
thin layers of gas or by investigating the ionizations at very] low pressures. Owing 

* *Comptes Rendus,' vol, 123, p. 878. 
t *Phil. Mag.,' January, 1899, p. 136. 



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TRAVERSED BY CATHODE RAYS. 73 

to mechanical difficulties, however, the former method is generally impracticable, 
while the action of the applied electric field in breaking down the insulation of the 
gas precludes the use of the latter artifice. 

It is then open to measure the ionizations produced by rays, traversing layers of 
gas of considerable thickness. But before any relation connecting ionizations and 
pressures can be deduced from such measurements, it is necessary to have definite 
information regarding the absorptive powers of the gases at different pressures, and 
to know exactly the form and dimensions of the region from which the ions are 
drawn. 

Although the absorption laws for cathode rays have been fully developed by 
Lenard, and are quite definite and clear, it is scarcely possible to define even 
approximately the region in the ionizing chambers (fig. 5) from which the ions go to 
make up the saturation current. 

On this account a direct verification of the proportionality law is not possible ; 
but, as already pointed out, the results of the experiments described in Section X. 
strongly support the conclusion that, in the case of a gas subjected to increasing 
pressure, the ionizations produced by rays of constant intensity bear the same ratio 
to each other as the coefficients of absorption corresponding to these pressures. 

If, then, the ionization in a gas varies with the pressure, it follows at once that if 
rays of the same intensity were allowed to traverse thin layers of different gases at a 
constant pressure, the ionizations produced would be directly proportional to the 
densities of these gases. 

Take, for example, carbon dioxide and air. It has been shown that the ionization 
produced in carbon dioxide at a pressure of 5047 millims. of mercury is the same 
as that produced in air at 7727 millims. by rays of the same intensity. 

According to the proportion law the ionization produced by these same rays in 
CO2 at 772 7 would then be just r53 times that obtained at the lower pressure ; 
that is, with rays of the same intensity the ionizations in carbon dioxide and in air 
would be to each other as 1*53 to 1 when these gases were subjected to the same 
pressure. 

A similar conclusion may be deduced from a consideration of the other gases 
examined. Hence, on this view, the relative ionizations produced by rays of 
constant intensity in a series of gases subjected to the same pressure would be 
expressed by the numbers which under these circumstances give their relative 
densities. 

These numbers are given for the gases examined in Column I., Table X., while in 
Column II. are given the values found by J. J. Thomson* for the relative ionizations 
produced by Rontgen rays of constant intensity in the same gases. 



* *Proc. Camb. Phil. Soc.,* vol. 10, Part I., p. 12. 
VOL. CXCV. — A. L 



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74 



MR. J. C, McLENNAU on electrical CONDUCTIVITY IN GASES 



Table X. 





Column I. 


' Column XL 


Gases examined. 


Densities (shown above 

to be proportional to 

ionization by cathode 

rays), air = 1. 


Ionization by 

Rontgen rays. 

Ionization of air taken 

as unity. 


Air 

Oxygen 

Nitrogen .... 
Carbon dioxide . . 
Hydrogen .... 
Nitrous oxide. . . 


1-00 1-00 
1-106 M 
•97 -89 
1-53 j 1-4 

•069 •as 

1-52 1-4.7 



The numbers, with the exception of those for hydrogen, present an agreement 
which is very striking, and show that although the two forms of radiation are so very 
different, still the products of their action upon the gases cited are practically the 



same. 



While the difference in the nimibers for hydrogen is very large, there seems to be 
some doubt as to the proper value to be assigned to the conductivity produced by 
Rontgen rays in this gas. The conductivities under Rontgen rays in the gases 
named have been measured by a number of experimenters, and while their values for 
the other gases differ but little, a very wide divergence exists in their numbers for 
hydrogen. Rutherford* gives the value '5, while PERRiNf bas obtained the 
number '026 by a method entirely different from that of any of the others. 

Though we have been thus led to conclude that the density of a gas should 
determine its conductivity under cathode rays, strong evidence exists against adopt- 
ing any such general conclusion regarding the conductivity produced by Rontgen 
rays, notwithstanding the general agreement indicated above for the gases cited. 

With such gases as HCl, Cl^, SO2, and HgS, J. J. Thomson, Rutherford, and 
Perrin have found the conductivities given in Table XI. 

From an examination of these values and a comparison with those of Table X., it 
is evident that it is quite impossible to deduce any such relation between the densities 
of the gases and their conductivities under this radiation. 



* * Phil. Mag.,' April, 1897, p. 254 

t * Th^se pr^ent^ k la Faculty des Sciences de Paris,* 1897, p. 46. 



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TRAVERSED BY CATHODE RAYS. 75 

Table XL — Conductivity under Rontgen Rays. 



Gas. Deiisitv. 



Measurcil by 

Pkrrin. 



J. J. Thomson. Eithkrford. 

I 



• HCl 1-25 .8-9 11 

SO., : 2-23 6*4 4 

i Cl.> 2-45 17-4 : 18 

, U^S 119 CO 6 



8 : 

G I 



Although the laws of ionization and absorption for cathode rays are clearly defined 
by these results, it is difficult to apply them in practice to the direct calculation of 
the relative ionizations in any particular experiment. 

Take, for example, the case of a pencil of pai*allel rays, 1 sq. centim. in cross 
section, traversing air at a pressure p. 

Let q = the rate at which ions are produced in 1 cub. centim. of air at unit 

pressure by cathode rays of unit intensity 
and \) = the coefficient of absorption of air for unit pressure. 

Consider then the ionization between two planes distant x and x + c/x, from the 
source of the rays. 

If I denotes the original intensity of the rays, I . e"**^ will represent their 
intensity at a distance x, and p.q .1. c'^^'^'dx wiU then represent the total number of 
ions produced between these two planes in one second. 

Imagine now a saturating electric field applied at right angles to the rays and 
confined between the limits r and r -{• d. 

The value of the total saturation current obtained with this field would then be 

fr+d 
p.q.l. e'^^dxy 

or i = ^.e-^'-(l-^e-^^) (1), 

At) 

where pX^ is replaced by the quantity X, whose values for different i)ressuro8 are 
given in Table IX. 

If the air traversed be now subjected to diminishing pressures, the saturation 
current will assume different values and will reach a maximum when 

i.e., {r+ d)e-'"'^r = 0, 

.- = '• + '' (2). 

L 2 



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76 MR. J. c. Mclennan on electrical conductivity in gases 

An experiment somewhat analogous to this is described, in Section IX. The 
apparatus used is shown in figs. 4 and 5. The diameters of the electrodes C and D 
were each about 1 centim. and, as already stated, the distance between the window 
and the centre of each of the chambers was about 2 centims. 

By applying the equation (2) to this experiment, and taking r = 15 centims.* and 
d = I centim., it follows that the saturation current would be a maximum when 

€^= 1-66 ... 
or X = '5. 

From Lenard's values, Table IX., it will be seen that this value corresponds 
approximately to a pressure of about 120 millims. of mercury. The observed results, 
however, Table II. and fig. 6, indicate a maximum of about 75 centims. Further, 
the calculated values of the current from equation (1) exhibit a more rapid rise than 
that actually observed. 

But the difierence in the results is not surprising. The field within the receiver 
was far from uniform, being disturbed by the proximity of the walls of the chamber. 
The presence of the narrow tube through which the rays were conducted into the 
receiver also produced irregularities. On this account it was impossible to define, 
even approximately, the region from which the saturation current was drawn. 
Moreover, the actual paths of the rays, as Lenard has pointed out, are largely 
influenced by the pressure of the gas traversed. Even at best, then, the calculated 
results can scarcely be regarded as more than a rough approximation. 



12. Summary of Results, 

1. The conductivity impressed upon a gas by cathode rays is similar to that 
produced by Rontgen and uranium rays, and can be fully explained on the hypothesis 
that positive and negative ions are produced by the radiation throughout the volume 
of the gas traversed. 

2. When cathode rays are allowed to fall upon insulated metallic . conductors 
surrounded by air at atmospheric pressure, 

(a.) such conductors if initially uncharged gain a small limiting negative charge, 

(6.) positive charges are completely dissipated, 

(c.) negative charges drop to a small limiting value, 

(o?.) the loss of charge is due to the action of the ionized air surrounding the 

conductor, and the value of tlie limiting negative charge is determined by 

the extent of the conduction in this air. 

3. The ionization produced in a gas by rays coming from the aluminium window 
in a Lenard discharge tube is due to cathode rays and not to Rontgen raya 



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TRAVERSED BY CATHODE RAYS. 77 

4. Lenard's results obtained by fluoroscopic methods on the absorption of cathode 
rays are confirmed by a study of the ionization these rays produce in gases. 

5. When cathode rays of a given strength are passed through a gas, the number 
of ions produced in 1 cub. centim. depends only upon the density of the gas, and is 
independent of its chemical composition. 

6. With rays of constant intensity the ionization in any particular gas varies 
directly with the pressure to which it is subjected. 

7. The relative ionizations produced by cathode rays of constant intensity in air, 
oxygen, nitrogen, carbon dioxide, hydrogen, and nitrous oxide, at the same pressure, 
are expressed by the numbers which represent their densities. 

8. With cathode rays, just as with Rontgen rays, the number of ions produced in 
a gas bears a definite ratio to the amount of the radiant energy absorbed. 

I gladly avail myself of this opportunity to record my grateful sense of the never 
failing encouragement and assistance received from Professor J. J. Thomson. 



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[ 79 ] 



III. Mathematical Contributions to the Theory of Evolution. — VIII. On the Inheri- 
tance of Characters not capable of Exact Quantitative Measurement. — Part I. 
Introductory. Part II. On the Inheritance of Coat-colour in Horses. Part III. 
On the Inhentance of Eye-colour in Man. 

By Karl Pearson, F.R.S., ivitk the assistance of Alice Lee, D.Sc.y University 

College^ London. 

Received August 5, — Read November 16, 1899 ; withdrawn, rewritten, and again received 

March 5, 1900. 

Contents. 

Pakt I.— Introductory. 

Page 

§ 1. General Nature of the Problem and Assumptions upon which it can be solved 80 

§ 2. Determination of the Mean Value of the Characters and the Ratio of the Variability of 

Correlated Characters 81 

§ 3. Determination of the Probable Errors and Error Correlations of all the quantities involved . 82 
§ 4. On the Construction of Normal Scales for Characters not capable a prim'i of exact 

quantitative Measurement 87 

§ 5. On Blended and Exclusive Inheritances 88 

Part II. — On Coat-colour Inheritance in tlie Tlwrougldned Horse. 

§ 6. On the Extraction and Reduction of the Data 92 

§ 7. On the Mean Coat-colour of Thoroughbred Horses 93 

§ 8. On the Relative Variability of Sex and Generation 94 

§ 9. On the Inheritance of Colour in Thoroughbred Horses 98 

(a) Direct Line, First Degree, {b) Direct Line, Second Degree. {c) Collateral 
Inheritance. General Conclusions. 

Part III. — On Eye-colour Inheritance in Man. 

§ 10. On the Extraction and Reduction of the Data 102 

§11. On the Mean Colour, having regard to Sex and Generation 104 

§ 12. On the Relative Varial)ility of Sex and Generation 109 

§13. On the Inheritance of Eye-colour in Man 113 

(a) Aflsortative Mating, (b) Collateral Heredity, First Degree, (c) Collateral 
Heredity,. Second Degree, (d) Direct Heredity, First Degree, (e) Direct 
Heredity, Second Degree. (/) On Exclusive Inheritance. 

§ 14. General Conclusions 119 

Appendix I. Tables of Coat-colour in Horses 122 

II. Tables of Eye-colour in Man 138 

Note I. Inheritance of Temper and Artistic Instinct 147 

Note 11. On the Correlation of Fertility and Eye-colour 148 

VOL. CXCV. — A 264. 29.10.1900 



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80 PROFESSOR K. PEARSON AND DR. A. LEE ON 

NOTE. 

Thifl memoir was originally presented to the Society on August 5, 1899, and read on November 16, 
1899. In working out by the same theory the coefficients of inheritance for Basset Hounds, Mr. Leslie 
Bramley-Moore discovered that the method adopted was not exact enough in its process of propor- 
tioning. Accordingly, with the assistance of Mr. L. N. G. Filon, we immensely developed the theory, so 
that it was necessary to rewrite the theoretical part of the original memoir. This has been carried out in 
Part VIL of this series. The present memoir consists substantially of the portions of the original 
memoir relating to the inheritance of coat-colour in Horses and eye-colour in Man, with the numerical 
details and the resulting conclusions modified, so far as the extended theory rendered this necessary. In 
the very laborious work of reconstructing my original tables I have received the greatest possible assistance 
from Dr. Alice Lee, and I now wish to associate her name with mine on the memoir.* The memoir 
was at my request returned to me for revision after it had been accepted for the * Philosophical 
Transactions.* 



Part I. — Introductory. 

(1.) A CERTAIN number of characters in living forms are capable of easy observation, 
and thus are in themselves suitable for observation, but they do not admit of an 
exact quantitative measurement, or only admit of this with very great labour. The 
object of the present paper is to illustrate a method by which the correlation of such 
characters may be effectively dealt with in a considerable number of cases. The con- 
ditions requisite are the following : — 

(i.) The characters should admit of a quantitative order, although it may be 
impossible to give a numerical value to the character in any individual. 

Thus it is impossible at present to give a quantitative value to a brown, a bay, or a 
roan horse, but it is not impossible to put them in order of relative darkness of shade. 
Or, again, we see that a blue eye is lighter than a hazel one, although we cannot 
d priori determine their relative positions numerically on a quantitative scale. 

Even in the markings on the wings of butterflies or moths, where it might be 
indefinitely laborious to count the scales, some half dozen or dozen specimens may 
be taken to fix a quantitative order, and all other specimens may be grouped by 
inspection in the intervals so determined. 

We can even go a stage further and group men or beasts into simply two 
categories — ^light and dark, tall and short, dolichocephalic and brachycephalic — and 
so we might ascertain by the method adopted whether there is, for example, correla- 
tion between complexion and stature, or stature and cephalic index. 

(ii.) We assume that the characters are a function of some variable, which, if we 

* I have further to thank Mr. Leslie Bramley-Moore, Mr. L. N. G. Filon, M.A., Mr. W. R. 
Macdonell, M.A., LL.D. and Miss C. D. Fawcett, B.Sc, for much help in the arithmetic, often for 
laborious calculations by processes and on tables, which were none the less of service if they were 
afterwards discarded for others. To Mr. Bramley-Moore I owe the extraction and part of the 
arithmetical reduction of the horse-colour tables. 



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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 81 

could determine a quantitative scale, would give a distribution obeying — at any rate 
to a first approximation — the normal law of frequency. 

The whole of the theoretical investigations are given in a separate memoir, in 
which the method applied is illustrated by numerical examples taken from inheri- 
tance of eye-colour in man, of coat-colour in horses and dogs, and from other fields. 
We shall not therefore in this paper consider the processes involved, but we may 
make one or two remarks on the justification for their use. If we take a problem 
like that of coat-colour in horses, it is by no means difficult to construct an order of 
intensity of shade. The variable on which it depends may be the amount of a 
certain pigment in the hair, or the relative amounts of two pigments. Much the 
same applies to eye-colour. In both cases we may fail to obtain a true quantitative 
scale, but we may reasonably argue that, if we could find the quantity of pigment, 
we should be able to form a continuous curve of frequency. We make the assump- 
tion that this curve — to at any rate a first approximation — is a normal curve. Now 
if we take any line parallel to the axis of frequency and dividing the curve, we 
divide the total frequency into two classes, which, so long as there is a quantitative 
order of tint or colour, will have their relative frequency unchanged, however we, in 
our ignorance of the fundamental variable, distort its scale. For example, if we 
classify horses into bay and darker, chestnut and lighter, we have a division which is 
quite independent of the quantitative range we may give to black, brown, bay, 
chestnut, roan, grey, &c. 

Precisely the same thing occurs with eye-colour ; we classify into brown and darker, 
hazel and lighter, and the numbers in these classes will not change with the 
quantitative scale ultimately given to the various eye-tints. Our problem thus 
reduces to the following one : Given two classes of one variable, and two classes of a 
second variable correlated with it, deduce the value of the correlation. Classify sire 
and foal into bay and darker, chestnut and lighter ; mother and daughter into brown 
and darker, hazel and lighter, and then find the correlation due to inheritance 
between the coat-colour or eye-colour of these pairs of relations. The method of 
doing this is given in Memoir VII. of this series. Its legitimacy depends on the 
assumptions (i.) and (ii.) made above, which may I think be looked upon as 
justifiable approximations to the truth. 

Of course the probable error of the method is larger than we find it to be when cor- 
relation is determined from the product-moment. Its value varies with the inequality 
of the firequency in the two classes given by the arbitrary division. It will be 
least when we make that frequency as nearly equal as possible — a result which can 
often be approximately reached by a proper classification. In our present data the 
probable errors vary from about "02 to '04, values which by no means hinder us from 
drawing general conclusions, and which allow of quite satisfactory general resulta 

(2.) So far we have only spoken of the two classes, which are necessary if we 
merely want to determine the correlation. But if we wish to deal with relative 

VOL, CXCV. — A, M 



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82 



PEOPESSOR K. PEAESON AND DR. A. LEE ON 



variability we must have more than two classes. We have, in fact, in our tables 
preserved Mr. Galton's eight eye-colour classes and the seventeen classes under which 
the coat-colour of thoroughbred horses is classified in Wethbrby^s studbooks. Such a 
classification enables us at any rate approximately to ascertain relative variability, 
and, what is more, to reconstruct approximately the quantitative scale according to 
which the tints must be distributed in order that the frequency should be normal. 
For, in order to attain this result, we have to ascertain from a table of the areas of 
the normal curve the ratio of the length of the abscissa to the standard deviation which 
corresponds to any given increase of frequency. Let us suppose that three classes 
have been made — n^, n^, nj, represented by the areas of the normal curve in the 
accompanying diagram so marked. Let pi and p^ be the distances of the mean from 




Ox, x^ 

the two boundaries of n^ Here Pi may be negative, or p^ infinite, &c. Then if 
Aj = p^ja-y A3 = pj(r^ we find at once, if N = total frequency, 

"^^ = viiy""^ (>■)• 

"■'r-'" = a/!D-"<^ (-)• 

Now the integrals on the right are tabulated, and thus, since the left-hand side is 
a known numerical quantity, it follows that pja and Ps/o-, and accordingly the range 
(Ps ~V\)I^ ^^ *^® (AsiSR in terms of the standard deviation, are fully determined. 

Thus, if € be the range on the scale of tint or colour of the group of which the 
observed frequency is ng, we have € = j^g — p^, and thus c/cr = q say, is known. 
For a second series c/cr' = q\ Hence a-ja = q/q, and accordingly the ratio of the 
variabilities of the two series is determined. 

Again, the ratio pj{pz — Pi) enables us to find the position of the mean in terms 
of the range on the scale occupied by the tint corresponding to the frequency n^. 
As a rule we shall take this tint to be that in which the mean actually lies, in which 
case we shall have pjipz + P\) as determining the ratio in which the mean divides 
the true quantitative range of this particular tint. 

(3.) Let 7; = p^{pz-Pi) = h/{K-K) (iii-)> 

^ = cr/ar' = (V-V)/(^3-^) (iv.)- 

It remains to find the probable errors of these quantities. 



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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 83 

Suppose Sj. to be the standard deviation for the errors in a quantity a?, and R,y 
the correlation coefficient for errors in two quantities x and y. 
Further let 

H^;^'-" W' 

where subscripts and dashes may be attached to H to correspond to like distin- 
guishing marks attached to h. 
Since 

2^1-^ _ 1 te-i^dx hi ) 

2N -v/2^Jo*' "^ ^^^-'^ 

we have at once Snj =: NHjS^i, 

and S,. = 2«y(NH0 (vU.). 

Similarly, Sn^ = — NH38A3, whence : 

24. = W(NH3) (viii.). 

Further, we have 2«.2>i,R*.«, = - S,,2..R«.,y(N«HiH3) (ix.) ; 

but, as is shewn in Part VII., § 4, 

, __ Wi(N-«A ^ ss _ n,(N-n,) , . 

MA..= -^ • . . (^.). 

Thus we find 

Probable error of ^1 = *67449Si, 

_ -67449 1_ A(N-n,) , •• . 

~ v/N Hi V N2~~ ^^"•^• 

Probable error of A3 = 'T/W'W A/ "' n~ (xiii.). 

S 

Correlation in errors in hi and A3, or R|^^, is given by 

2*.S^RiA=l^ (^^^-^ 

Let u = A3 — Aj, tt' = A3' — A/ be the ratio to the respective standard deviations 
of the ranges corresponding to the groups n^ and rij'. Then 

~ N»l Hi« "^ H,« HiH,J ' 

whence, if i/ be a proportional frequency = n/N, we readily find 

Prol»bleerrorofu = -^''{^+i-(^_ + ^7 . . . (xv,). 
Rotable «ror of u'=:^'{|l+|l-(i + ^,)*}'. . . (.vi.). 

M 2 



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84 



PROFESSOR K. PEARSON AND DR. A. LEE ON 



I now proceed to determine the correlation in the errors made in determining the 
ranges corresponding to any two classes of any two variables which are correlated. 
For this purpose let the frequency correlation table be dressed as follows, in the 
diagram below. 





kA-j>i< — 


-h.-k- 


» 


i 




Axis of X variable, g ': 


'/ 




*. 


TotAL 


^ 














§• 
















/77„ 




m^i 




"^Sl 


n', 


\s-0- 


M 








^ 


\ 


^%y, 




> 




q'' 




If- 


m,g 




m^g 




fUss 


"i 


> 














1 

1 




T 




s 






5& 


V^ 








HsHJ) 






fn,s 




/»« 




Mas 


r>^ 


ToCclLs 


", 




n. 




n^ 


N 



Here m,y denotes the frequency of individuals common to the two classes n,- and 
riy Let M(/ denote its " conjugate," or all the frequency which appears in neither Ui 
nor v!j ; then 

N = My + n, + n'j — m^ (xviL). 

As before, we have 



J ^ n i(N — n,) 

N 






2_ »t'XN-«0) 



N 



g _ 'MyCN-^fy) ^ 2 _ mi,(N - V/iy) 

'^ — N ' ^^ — N 



(xviii.). 
(xix.). 



Further, since m-^ and M(; are mutually exclusive, we have 



Mj/ZWy 



From (xvii.) we have for small variations 



(XX.). 



Hence 



2S^X,R.^,= V+V-S-.*- V-22.,tH,RM^ . . . (xxi). 



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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 85 

Substituting the values given above we find, after some reductions, 

2„.2«..K^,;,^ = —^-^ ' (xxu.) 

This result, which is extremely simple in form, gives the correlation in errors made 
in determining the frequencies in any two classes whatever of any two correlated 
variables. 

I next proceed to find the correlation between errors in u and u\ the ratio of the 
ranges occupied by any two classes to their respective standard deviations. 

We have 

8n^ = — NH38A3. 
Hence S(A, - *,) = ^ + I'd; - i). 

Similarly 8(V - A,-) = ^. + f (g. - |.). 

Multiply the first by the second, and summing as usual for all possible errors, we 
have, by using (xxii.) 

^^p _ 1 JNmaa - 71^71^' . Nm^g -' n^n^' / 1 1 \ 'Sm^ - n^\ / 1 1\ 

z«z,.Xi,^ - N 1 N^HiH/ " ■•" N»Hi VHi' sj ""^ N^H/ \Ei, ~ hJ 

Collecting the like H's we find, after very considerable reductions, 

. . . . (xxiiL), 
, Z.2,^n,,. - jj I jj^jj^, -h jj^jj^, -f jj^g^, -I- jj^jj^, I . . vxxui. ;. 



or, 



where /ty = wi^/N = proportional frequency. 

A glance at our diagram on the previous page of the correlation table divided into 
nine classes, shows at once the symmetrical formation of this result. By writing at 
the points P, Q, S, and T, the ordinate there of the normal surface, on the supposition 
of no correlation and N = 1, the construction of the result is still more clearly 
brought out. 

We are now in a position to determine the probable errors of 17 and ^ We have 



8,,= 



__ h^Bh^ — AjS/ij 



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86 PROFESSOR K. PEARSON AND DR. A. LEE ON 

Hence 

- «*N 1 (AiH,)«Na "^ (A8H3)'N2 (A,Hi)(A2H,)N2i ' 
Or, Probable error of t) 

where u is the range A3 — h^, and v^ and 1^3 are the proportional frequencies, as before. 
Care must be taken, if the class n^ cover, as it usually will in our present investiga- 
tions, the mean, to put h^ negative within the radical. In other words, for a class 
covering the mean we have 

Probable error of rj 

- '67449 h,K r_?L . _-s_ _ fjH Ys^yV /xxv^ 

Lastly we have 

u* u \u' tt / * 

*f ~ »» \ V,'* "^ tt« ««' J • 

Thus : Probable error of ^ 

= -67«.i{|'+^--^}' (.xvi.), 

where we have by (xv.), (xvi.), and (xxiii."*) 

2 8-1 Jiii . jv _/< . ivyi 

V < p _ i J /*ii - "i^i' , M »8 - ^sV , /*i8 - yiV _i_ /^i r "V!!'! 

*"^"' """Nl H,Hi' ^"HsH,' "•■ H1H3' +'H3Hi' /' 

where, as before, fi'a and v'a represent proportional frequencies. 

In the following investigations on coat-colour, and eye-colour inheritance I have 
not thought it needful to give in every one of the thirty-six relationships dealt with 
the probable errors of the means, ratio of variabilities, and the coefficients of inheri- 
temce (»;, 4> and r). The arithmetical labour would have been too great, for the 



or, 
Hence 



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MATHEMATICAL CONTEIBUTIONS TO THE THEORY OP EVOLUTION. 



87 



expressions as given above are somewhat complex. It is, however, necessary to keep 
the approximate values of these probable errors in view, and, as our results classify 
themselves easily into groups for which our data, as well as the intensity of heredity, 
are approximately the same, one series of these errors has been found for each group. 

(4.) If we have ground for our assumption that the variable at the basis of our 
tint classification can be so selected as to give a normal distribution, we may deter- 
mine the relative lengths on the scale of that variable occupied by each tint or shade. 

Thus if (Ti be the standard deviation of the variable for male eye-colour, o-g for 
female eye-colour, I measured the range on the scale in terms of o-j and a^ for 
Mr. Galton's eight eye-colour tints for 3000 cases of male and 3000 cases of female 
eye-colour. I found the spaces occupied on the unknown scale to be as follows : — 



No. 


Tint. 


Eange in terms of 0-2. 


Range in terms of o-i. 


1 
2 
3 
4 
5 
6 
7 
8 


Light blue . * 
Blue, dark blue 
Grey, blue-green 
Dark grey, hazel 
Light brown . 
Brown . . . 
Dark brown. . 
Very dark brown. 


black! 


00 
1-39276 
•73468 
•40027 
•03893 
•43679 
•84161 

00 


00 

1-34918 
•77596 
•41992 
•00856 
•35895 
•64167 
00 



These results are not so regular as we might have hoped for, on the assumption 
that the ratio of a-i/a-t^ would be the same from whatever part of the scale it be 
determined. The general conclusion, however, would be that a-^ is slightly larger 
than 0*2, which is confirmed by other investigations. Actually a tint may be rather 
vaguely described, and where the data were obtained by untrained observers without 
the assistance of a plate of eye-colours, a good deal of rather rough classification is 
likely to have taken place. I do not think it would be safe to go further than stating 
that on the quantitative colour scale the tints as described must occupy spaces in 
about the following proportions : — 



Light 
Blue. 


Blue, 
Dark Blue. 


Grey, 
Blue-Green. 


Dark Grey, 
Hazel. 


Light 
Brown. 


Brown. 


Dark 
Brown. 


Very dark Brown, 
Black. 


00 


1-37 


•75 


•41 


•02 


•40 


•74 


00 



Taking 2000 colts and 2000 fillies, the standard deviations being a-i and 0-3 respec- 
tively, I have worked out the coat-colour ranges in terms of 0-3 and cr, for each of 
the sixteen colours* occurring in the records. We have the following results ; — 

* See p. 92, below. 



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88 



PROFESSOR K. PEARSON AND DR. A. LEE ON 



Tint. 


Range in 0-2. 


Range in a-^. 


Range in o-j. 


Range in o-j. 


1 


00 


ao 


9 


•00000 


•00000 


2 


•12683 


•10768 


10 


1-96956 


2-01658 


3 


•00000 


•03313 


11 


•00000 


•00000 


4 


•91747 


11 1055 


12 


•02490 


•00000 


5 


•00000 


•00352 


13 


•00000 


•00000 


6 


•11059 


•10451 


14 


•00000 


•00000 


7 


1 •34684 


1 •27688 


15 


•00000 


•00000 


8 


•00000 


•00000 


16 


00 


00 



Here again it seems to me that the most we can safely do is to consider that on a 
suitable scale the relative lengths occupied by the classes of coat-colours recognised 
by thoroughbred horse breeders would be somewhat as follows : — 



bl. 


bl./hr. 


br./bl. 


br. 


br./b. 


b./br. b. 


b./ch. 


ch./b. 


ch. 


ch./ro. ro./ch. 

i 


ro. 


ro/gr. 


gr/ro. 


g^. 


' 00 


•12 


•02 


roi 


•00 


•11 1^31 


•00 ' •oo 


1-99 


•00 


•01 


•00 


•00 


•00 


00 

1 



The reader must carefully bear in mind that these represent scale-lengths occupied 
by the coat-colour and not the frequency of horses of these individual coat-colours. 
What we are to understand is this : that if eye-colour in man and coat-colour in 
horses were measured on such quantitative scales as we have given in skeleton, then 
the distribution of the frequency of the several colours would be very approximately 
normal. The actual skeleton scales are represented in the accompanying diagram, 
which puts them at once before the eye. 



br/bC. 



T 



Normal Scale 0(f Colour Ranges in Thoroughbred Horoes. 

tibr. ch/h. 






BLsLck 



Brown 



Bay 



Chestnut 



smn 



tr/b. 



^m 



chM'Si^. 



Normal Scale of Eye Colour Ranges in Man. 



am^^^"^' 



Brtmr, %S^ 




Grey 
BLue-6reen 



Odrk blue. 
Blue 



LighC 
BlSe 



(5.) It is necessary here to draw attention to a distinction of some importance in 
heredity, namely, that between blended and exclusive inheritance. In my treatment 
of the law of ancestral heredity,* it is assumed that we have to deal with a quanti- 
tatively measurable character, and that the ancestry contribute to the quantity of 
this character in certain proportions which on the average are fixed and follow certain 
definite numerical laws. Such an inheritance is blended inheritance. But another 

* *Ro7/Soc. Proc.,' vol. 62, p. 386. 



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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 89 

type of inheritance is possible. We may have one progenitor, prepotent over all others 
and absorbing all their shares, who hands down to the of&pring not a proportion of 
his character, but the whole of it without blend. If this progenitor is a parent we 
have exclusive inheritance, if a higher ancestor a case of reversion. I have dealt at 
some length with this type of inheritance imder the title of the Law of Reversion 
in another paper.* We must consider in outline the main features of such inheritance, 
for the cases of eye-colour in man and coat-colour in the horse approximate more closely 
to the numerical values required by it, than to those indicated by the law of ancestral 
heredity. The chief feature of exclusive inheritance is the absolute prepotency of 
one parent with regard to some organ or character. It need not always be the 
parent of the same sex, or the same parent throughout the same family. Some 
offspring may take absolutely after one, others after another parent for this or that 
organ or character only. I believe Mr. Galton first drew attention, in his * Natural 
Inheritance' (p. 139), to this exclusive or, as he terms it, alternative heritage 
in eye-colour. In going through his data again I have been extremely impressed by 
it ; even those cases in which children might be described as a blend, rare as they are, 
are quite possibly the result of reversion rather than blending. If we suppose exclu- 
sive inheritance to be absolute, and there to be no blending or reversion, it is not hard 
to determine the laws of inheritance. Supposing the population stable, one-half the 
oflGspring of parentages with one parent of given eye-colour would be identical with 
that parent in eye-colour, the other half would regress to the general population 
mean, i.e., the mean eye-colour of all parents. Hence, taken as a whole, the regression 
of children on the parent would be '5. In the case of the grandparent the regression 
would be '25 ; of a great grandparent '125, and so on. With an imcle a quarter of 
the oi&pring of his brother will be identical in eye-colour with him, the other three- 
quarters will regress to the population mean, thus the regression will be '25. If we 
have n brethren in a family, and take all possible pairs of fraternal relations out of it, 
there will be \n{n — 1) such pairs ; ^ brothers will have the same eye-colour that of 
one parent, the other \n brother that of the other parent. Hence selecting any one 
brother, \n — 1 would have his eye-colour, and on the average \n would have 
regressed to the mean of the general population. In other words, the coefficient of 
regression would be {\n — l)/(Jn -^ 1 + ^) = (^n — l)/(w — 1). 

Accordingly 



«= 3 


Begressiou 


= -25 


n = 4 




= -3333 


«= 4-7 




= -3649 


» = 5 




= -375 


n = 5-3 




= -3833 


n = 6 




= '4 


n = 00 




= '5 



♦ *Koy. JSoc. Proc./ vol. 66, pp. 140 d setj. 
VOL. CXCV. — A. N 



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90 PROFESSOR K. PEARSON AND DR. A. LEE ON 

It will at first appear, therefore, that the fraternal regression with the size of 
families actually occurring will vary from '35 to '4. 

To some extent these values would be modified by assortative mating, which 
actually exists in the case of eye-colour. The correlations between parent and 
offspring and between brothers would both be slightly increased. Thus if p be the 
coefficient of assortative mating, Ry the fraternal correlation with and Vf without 
assortative mating, and r the coefficient for parent and offspring,* 

If we put Vf = '36, ?' = '5, /o = '1, we find 

R/ = -39. 

Thus we see that the regression or correlation for fraternal inheritance in the case 
of exclusive inheritance could not, with the average size of families, be very far from 
'4 of blended inheritance. 

A further source which can modify immensely, however, the fraternal correlation is 
the prepotency of one or other parent, not universally, but within the individual family. 
In the extreme case all the offspring might be alike in each individual family. Thus 
fraternal correlation might be perfect although parental correlation were no greater 
than -5. Hence, where for small families we get a fraternal correlation greater than 
•4 to '5, it is highly probable that there exists either a sex prepotency (in this case, 
one of the parental correlations will be considerably greater than the other) or an 
individual prepotency (in which case the parental correlations based on the average 
may be equal). We shall see that fraternal correlations occur greater than '5 in our 
present investigations. I have dealt with these points in my Memoir on the * Law of 
Reversion,'! and also in the second edition of the ' Grammar of Science. 'J 

Another point also deserves notice, namely, that with the series '5, '25, '125, &c., 
for the ancestral coefficients in the direct line, the theorems proved in my Memoir on 
Regression, Heredity, and Panmixia§ for the series of coefficients ?% r^, r^ . . . exactly 
hold. That is to say, if we have absolutely exclusive inheritance, the partial regres- 
sion coefficients for direct ancestry are all zero except in the case of the parents. 
This it will be observed is not in agreement with Mr. Galton's views as expressed in 
Chapter VIII. of the * Natural Inheritance.' But I do not see how it is possible to 
treat exclusive inheritance on the hypothesis that the parental regression is about '3.11 
Actual investigation shows that for this very character, i.e., eye-colour, it is nearer '5. 
If we take Table XIX. of Mr. Galton's appendix, and make the following groups, both 

* This is shown in a paper not yet published on the inj9uence of selection on correlation. 

+ *Roy. Soc. Proc./ vol. 66, pp. UO et seq„ 

t " On Prepotency," p. 459 ; " On Exclusive Inheritance," p. 486. 

§ ^Phil. Trans.,' A, vol. 161, p. 302, etc. 

II Mr. Galton takes I throughout his arithmetic. 



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MATHEMATICAL C50NTRIBUTI0NS TO THE THEORY OF EVOLUTION. 



91 



parents light, one parent light and one medium, one light and one dark, we reach the 
following results : — 



Parents* eye-colour. 


Children, actual. 


Light-eyed children, calculated. 


Total. 


l-'o-Hy^i.: iJSJl 


Ancestral law with 

knowledge of parents 

and grandparents. 


Both light. . . . 
Light and medium . 
Light and dark . . 


355 
215 
211 


334 355 

170 161 
107 105 

1 


321 
160 
117 

1 



Here the exclusive inheritance leads us to misplace thirty-two and the -ancestral 
law thirty-three children. The evidence, therefore, of the correctness of the latter is 
hardly greater than that of the former. Indeed, if the former were modified for 
reversion, it would very possibly give better results than the latter. 

I am inclined accordingly to look upon eye-colour inheritance as an exclusive 
inheritance modified by reversion, and, to some extent, by assortative mating, rather 
than a mixture of exclusive inheritance with a slight amount of blending. In either 
case exclusive inheritance gives results like the above so closely in accord with the 
ancestral law that the latter might be supposed to hold. But, theoretically, I do not 
understand how the ancestral law is compatible with exclusive inheritance. 
Theoretically, we have values of parental, avuncular, and grand-parental correlation 
greater than the ancestral law would permit of, and these theoretical values are, on 
the whole, closer to observation, as we shall see in the sequel, than those given by the 
law of ancestral heredity. The following table gives the two systems : — 

Table I. — Theoretical Values of the Regression Coefiicients. 



Belationship. 


Blended inheritance, 
ancestral law. 


Exclusive inheritance, 
absolute, no reversion. 


Parent and offspring 

Qrandparent and offspring . . . 
Great-grandparent and offspring . 

Brethren 

Uncle and nephew 


•3 

•15 
•075 
•4 
•15 


•5 
•25 
•125 
•36 to -5* 
•25 



Now, if exclusive inheritance be modified by reversion or assortative mating, or if 
blended inheritance be modified by " taxation," t then we shall get values. different 

* This varies with the size of the family, 
t * Roy. Soc. Proc.,' vol. 62, p. 402. 
N 2 



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92 PROFESSOR K. PEARSON AND DR. A. LEE OK 

from the above, and possibly filling up the numerical gap between them. To this 
point I shall return when dealing with the observed values for eye-colour in man. 



Part II. — On Colour-Inheritance in Thoroughbred Racehorses. 

(6.) All the data were extracted from Weatherby's stud-books, the colours being 
those of the horses as yearlings. My first step was to form an order, not a quantita- 
tive scale, of horse-colours. With this end in view, the recorded colours were 
examined, and, including the arabs, the following seventeen colours were at first 
found : — 



1. Black (bl.). 

2. Black or brown (bl./br.). 

3. Brown or black (br./bl.). 

4. Brown (br.). 

5. Brown or bay (br./b.). 

6. Bay or brown (b./br.). 

7. Bay (b.). 

8. Bay or chestnut (b./ch.). 

9. Chestnut or bay (ch./b.). 



10. Chestnut (ch.). 

11. Chestnut or roan (ch./ro.). 

12. Roan or chestnut (ro./ch.). 

13. Boan (ro.). 

14. Roan or grey (ro./g.). 

15. Grey or roan (g./ro.). 

16. Grey(g.). 

17. White (w.). 



Now, if we take the alternative colours to mean that the first alternative is the 
prominent element, we see that these colours in use among breeders admit of only one 
arrangement from black to white. That is to say, that a continuous shade-change is 
actually in use for the colour-nomenclature of thoroughbred horses.* Thus without 
any hypothesis as to the quantitative relative values of bay or roan, we have an order 
which serves for all our present purposes. Following this order. Appendix I., Tables 
I. — XII., for the colour correlation of related pairs of horses was compiled by 
Mr. Leslie Bramley-Moore from the stud-books. When dealing with relationship 
m the ? line only, no weight has been given to fertility, as each mare has had only 
one foal attributed to it, or two in the case of fraternal correlation. In the case 
of the c? line, the colours of the older sires were harder to ascertain, and we did not 
obtain altogether more than 600 sire-colours. Thus one, two, or, in a few cases, three 
or four colts or fillies were taken from each sire. 

I shall now discuss the results which may be drawn from these tables for the theory 
of heredity, first placing in a single table all the numerical constants calculated from 
the data in Tables I. to XII. of Appendix I. 

^ Among the 6000--8000 horses dealt with only four were found with colours not entered in this scale, 
but these entries of bl./ch., br./ch., b./ro., in no way contradict the order of the scale, but merely show a 
rougher appreciation on the part of the nomendator, or possibly printers' or editor's errors. 



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MATHEMATICAL CONTBIBUTIONS TO THE THEORY OF EVOLUTION. 



93 



Table II. — Coat-colour Inheritance in Thoroughbred Horses. 



Pair of relatives. 


Division of 

bay range by 

the mean 


Ratio of 
variabilities. 


Coeffi- 
cients of 
correlar 
tion. 


Coefficients of 
regression. 


Num- 
ber of 
cases. 


X. 


y- 


Vx- 


Vv 




C = 

<ryl<rg. 


rxg. 


Rw 


Rv«. 


N. 


Sire 


Colt. . 
FiUy . 
Colt. . 
Filly . 


•6111 
•6061 
•5359 
•5565 


•5713 
•5719 
•6027 
•6051 


•8712 
•8298 
•9500 
•9036 


1-1478 
1-2051 
1^0526 
11067 


•4913 
•5422 
•4862 
•5668 


•4280 
•4499 
•4619 
•5122 


•5639 
•6534 
•5118 
•6273 


1300 
1050 I 
1000 ! 
1000 


Sire 

Dam 

Dam 


Maternal 
Maternal 


grandsire 
grandsire 


Colt. . 
Filly . 


•6583 
•6359 


•5867 
•6042 


•7030 
•7678 


r4225 
13024 


•3590 
•3116 


•2524 
•2392 


■5107 
•4058 


1000 
1000 


Colt . 
Colt . 
Filly . 
Filly . 
Filly . 
FiUy . 


'(Half 
(Whole 

(Half 
(Whole 

(Half 
(Whole 


Colt. . 
siblings) 

Colt, . 
siblings) 

Filly . 
siblings) 

Filly . 
siblings) 

Colt. . 
siblings) 

Colt. . 
siblings) 


•5908 
•5620 
•5665 
•5684 
•5633 
•5410 


•5908 
•5620 
•6665 
•5684 
•5865 
•5711 


1 

1 

1 

1 
•9607 
•9555 


1 

1 

1 

1 
1^0409 
1^0466 


•3551 
•6232 
•4265 
•6928 
•2834 
•6827 


•3551 
•6232 
•4265 
•6928 
•2723 
•5568 


•3551 
•6232 
•4265 
•6928 
•2960 
10466 


2000 
2000 
2000 
2000 
1000 
1000 



In this table R,y = r^^^cxlcy, 'Ryx = ^jryO-y/cr^r. Half-siblings* are those having the 
same dam, but different sires. Further, 17 is measured from the brown end of the bay 
range up to the mean. 

(7.) On the Mean Coat-Colour of Horses. — If our theory be correct, this colour 
will not differ much from the median colour, and we notice at once a secular change 
going on. We have the following order : — 

Maternal grandsire of colt t; = '6583 

Maternal grandsire of filly = '6359 

Sire of colt = -6111 

Sire of filly = 6061 

Colt (mean value of seven series) . . . = '5816 

Dam of colts t; = '5359 

Dam of fillies = -5565 

Fillies (mean value of seven series) . . = *575S 

* I have introduced this expression in my paper on "The Law of Reversion," *Roy. Soc. Proc./ vol. 66, 
as a convenient expression for a pair of offspring from same parents whatever be their sex. 



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94 PROFESSOR K. PEARSON AND DR. A. LEE ON 

Now the colours of all the horses are returned when they are foals, so that there is 
no question of any variation of colour with age, yet we notice that — 

(i.) The horse is lighter in colour than the mare. 

(ii.) If we go back two generations (grandsire) the horse is lighter than if we only 
go back one generation (sire), and the sires are again lighter than their colts. 
In other words, there seems a progressive change towards a darker coat. 

(iii.) On the other hand, the mares one generation back appear to be darker than 
their daughters. 

(iv.) The average sire of colts is lighter than the average sire of fiUies ; the average 
dam of colts is darker than the average dam of fillies. 

Now these conclusions seem to indicate that the older horse was lighter in coat, and 
the older mare darker in coat than either the colt or filly of to-day, and that there is a 
tendency in the thoroughbred racehorse of to-day to approach to an equality of colour 
in the two sexes, an equality which is a blend of the sensibly divergent sex-colour of 
the older generation. 

Whether this secular change is due to the " breeding out " of the influence of light 
Arabian sires, or to a tendency in the past to select light-coloured sires and dark- 
coloured mares for breeding, or to the fact that such coloured sires and mares are the 
most fertile, i.e., to an indirect effect of reproductive selection, is not so easy to 
determine. But what does appear certain is that the average thoroughbred is 
approaching to a blend between its male and female ancestry, which were sensibly 
divergent.* 

(8.) On the Relative Variability of Sex and Class in Horses. — The following table 
gives the length of the bay range in terms of the standard deviation for each class. 
If c represent this range, then in terms of the previous notation c = w x <r = ?/ X o^, 
and from these values of it and it' the ratio, C = <^/<^' of Table II. was calculated. 



* Mean of dams and sires of colts = -5735, «.€., i(*6111 + -5359). 
Mean of dams and sires of fillies = *5813, i.e,, ^(*6061 + -5565). 

These are curiously enough almost exactly equal to the mean values '5753 and '5816 obtained for fillies 
and colts. This inverse relationship is too close to the probable errors of the quantities under investiga- 
tion for real stress to Ijc laid on it, but it may still turn out to be suggestive. 



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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 



95 



III. — Table of Bay Eanges. 



Relative Pair. 


Bay Ranga 


Probable Error of Median. 


X. 


y- 


U X a-,. 


u' X a-y. 


X. 


y- 


Sire 

Sire 

Dam 

Dam 


Colt .... 
Filly .... 
Colt .... 
Filly .... 


l-46943<r« 
l-64075<ra, 
l-36645<r, 
l-38165<ra 


l-28019(ry 
l-36I49<ry 
l-298I9<ry 
l-24845(ry 


± 0160 
±•0159 
±•0196 
±0193 


± -0183 
± ^0192 
±•0206 
±•0214 


Maternal grandsire . 
Maternal grandsire . 


Colt .... 
Filly .... 


l-69694<r, 
1 -650210^, 


M9293<r„ 
l-26702<r„ 


±•0158 
±0162 


± ^0224 
±0211 


Colt 

(Half 

Colt 

(Whole 

Filly 

(Half 

Filly 

(Whole 

Filly 

(Half 

FUly 

(Whole 


Colt .... 
siblinin) 

Colt . . . . 
siblings) 

Filly . . 
siblings) 

Filly .... 
siblings) 

Colt .... 
siblings) 

Colt .... 
siblings) 


l-23953<ra, 
l-27688(r« 
l-39619(ra, 
l-34684<r, 
l-33479<r, 
1 -415010^, 


l-23953<ry 
l-27688<ry 
l-39619<ry 
l-34684<ry 
l-28229<ry 
l-35207(r„ 


±0153 
±•0148 
±0135 
±•0140 
±0202 
±•0189 


±•0153 
±0148 
±•0135 
±•0140 
±0208 
±0198 



To explain the last double column I note that Mr. Shbppabd has shown (* Phil. 
Trans./ A, voL 192, p. 134) that the probable error of the median equals 

•84535 o-/\/N. 
Hence in terms of the bay range we have 



probable error of median .q^kqk// /m\ 

length of bay ranee /\ v /• 



length of bay range 

I have found that this simple result gives a value close to but slightly larger than 
the probable error of the quantity tj (p. 82), from which I have determined the 
position of the mean in the bay range. It is much easier to calculate, but of course 
not so exact, as we take no accoimt of possible errors in the bay range itself or 
their correlation with errors in the median. I have accordingly tabulated its values 
in the last double colunm as a rough guide to the errors made in the numbers 
upon which the statements in the previous section depend. I shall return to the 
consideration of the probable errors below. Turning to columns 3 and 4 of our 
Table 11. , we can draw the following conclusions as to the variability of sex and 
class: — 

(a.) The Younger Generation is more Variable than the Old. — Thus, foals are more 
variable than their sires, and, looking at the expressions in Table III. for the bay range, 



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96 PROFESSOR K. PEARSON AND DR. A. LEE ON 

sii'es than grandsires. This is a rule I have now found true in a very great number 
of cases of inheritance. Parents are a fairly closely selected body of the general 
population, and so less variable than that population at large. This might appear 
pretty obvious in the case of thoroughbred horses when we are dealing with sires and 
grandsires. I have already pointed out that it was impossible to take 1000 to 1300 
colts or fillies with as many independent sires, the fashion in sires is too marked ; and 
of course the number of independent grandsires was still fewer.* But even in the 
case of dams, where we have taken as many independent dams as fillies, we see this 
reduction in variability in the older generation. As it also occurs with stature, &c., 
in man as well as with coat-colour in horses — in which latter case we expect artificial 
selection — it deserves special consideration. Without weighting with fertility, there 
exists a selection of the individuals destined to be parents in each generation. We 
have to ask whether the change in mean and variability from parent to oflGspring in 
each generation is secular or periodic, or to what extent it is partly one and partly 
the other. The importance of settling this point is very great ; it concerns the 
stability of races. So far as my fairly numerous series of measurements yet go, 
I cannot say that a " stable population " has definitely shown itself; the characters of 
each race when measured for two generations seem to vary both in mean and 
standard deviation. Palseontologists tell us of species that have remained stable for 
thousands of years, but this is a judgment hitherto based on a qualitative apprecia- 
tion. A quantitative comparison of the means, variabilities, and correlations of some 
living species in its present and its fossil representatives would be of the greatest 
interest and value. For myself, I must confess that my nmnerical investigations so 
far tend to impress me with the unstable character of most populations. 

(6.) There is fairly good evidence that the horse is more variable than the 7na/re in 
coat-colour. It would be idle to argue fi:om the first four results of Table III. that 
the mare is more variable than the horse, in that these results show the dam to be 
more variable than the sire. For, as we have shown, the process of breeding and our 
method of extracting the data tend to produce a much more intense selection of sires 
than of dams. But if we compare the mean bay range in terms of the standard 
deviation of colts for our seven series of colts with that for the seven series of fillies 
in Table III., we find for the first 1*27458 <r<. and for the second 1 '33854 c/. Hence 
we are justified in concluding that a-c is greater than oy; In fact in only one case out 
of the seven does the series of fillies give a less variability than the corresponding 
series of colts, i.e., colts corresponding to dams are somewhat less variable than fillies 
corresponding to dams. It must, however, be remembered that this conclusion is 
based upon the coat-colour of the animals recorded as yearling foals, t Thus, the coat- 

* For some account of the extent of in and in breeding in the thoroughbred horse, see my memoir on 
" Reproductive Selection," • Phil. Trans.,' A, vol. 192, p. 267 et seq, 

t The reader must always bear in mind that when we speak of the variability of colour in sire or 
dam, <^c., it means the varialiility of this class when they were yearlings. 



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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 



97 



colour may change both in intensity and variability with age, much as variability in 
stature changes with children from birth to adult life. 

(c.) As a more or less natural result of (6) it follows that any • group, male or 
female, having nutle relatives is more variable than the same group with female 
relatives. Thus sires of colts are more variable than sires of fillies ; fillies half-sisters 
to colts are more variable than fillies half-sisters to fillies, &c. But out of the nine 
cases provided by our data there are three exceptions to the rule, and perhaps not 
much stress can be laid on it, at iany rate in the above form. It would appear that 
males, relatives of males, are sensibly more variable than males relatives of females. 
The bay ranges are 1*3926 </ and 1*4447 a- respectively, which indicates that the 
average cr' is larger than a-. But if we treat the groups of females alone, we find for 
females with male relatives the bay range = 1*3694 <r, and for females with female 
relatives 1*3433 <t\ which indicates that the latter are more variable. The difference 
is, however, not very sensible. Possibly the rule is simply that extremes tend to 
produce their own sex, but our data are not sufficient for reaUy definite conclusions on 
the point. 

In order that we may have a fair appreciation of the probable errors of the 
quantities involved and the weight that is to be laid upon their differences, I place 
here a table* of the probable errors of ij, of £ = Cx/a-y and of r,^ for t3rpical cases. 

IV.— Table of Probable Errors. 



Belations. 


V' 


Vy 


C 


u 


u' 


ray 


Sire and Filly . . 
Grandsire and Colt 
Colt and Colt . . 
(Whole siblings) 
Filly and Colt . . 
(Half siblings) 


•0U3 
•0143 
•0186 

•0179 


•0170 
•0199 
•0186 

•0185 


•0243 
•0237 

•0315 


•0363 
•0385 
•0328 

•0335 


•0330 
•0319 
•0328 

•0328 


•0288 
•0333 
•0259 

■0363 



It will be seen from this table that the probable error in 17 is about 3 per cent., in 
C about 2 to 4 per cent., in u about 2 to 2*5 per cent., and the values of r about *03, 
growing somewhat larger as r grows smaller. The probable errors are thus some- 
what larger than those which we obtain by the old processes when the characters are 
capable of quantitative measurement, but they are not so large as to seriously affect 
the use of the new processes in biological investigations. As we have already 
indicated, the probable errors of the tj'b may be roughly judged by Mr. Sheppard's 
formula for the median (p. 95). 

It will be seen that the differences in the rfa and fs of Table II., or the u'q of 

* I have to thank Mr. W. R. Macponell for friendly aid in the somewhat laborious arithmetic 
involved in calculating these probable errors. 

VOL. CXCV. — A. O 



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98 PROFESSOR K. PEARSON AND DR. A. LEE ON 

Table III., are as a rule larger than the probable errors of the diflferences, sometimes 
several times larger. Yet in some cases they are not such large multiples of the 
probable errors of the diflferences that we can aiSbrd to lay great stress on the 
divergence of ly or £ or tt in a pair of special cases. We must lay weight rather on 
the general tendency of the tables when all the series are taken together. Thus, 
while we may have small doubt about the correctness of (i.) of § 7 or (6) of § 8, we 
should look upon (iv.) of § 7 as an important suggestion which deserves serious 
consideration rather than a demonstrated law. The same again holds good for (c) of 
§ 8. It is because of their suggestiveness that they are here included. That a 
differential fertility or an individualisation in the sex of offspring should be corre- 
lated with colour, would, if proved, be a fact of very considerable interest. It would 
again emphasise the important part which genetic selection plays in the modification of 
characters.* A priori it is not more unreasonable to expect coat-colour in horses than 
to suppose hair-colour in men to be correlated with fertility. But the fertility of man 
does seem to vary from the light to the dark races. The special feature of interest 
here, however, is that a different colour in the two sexes appears to influence the 
preponderance of one or other sex in the offspring. It would be an interesting 
inquiry to determine whether the sex-ratio in the oflfepring of " mixed marriages " 
varies when the races of the two parents are interchanged. 

(9.) On the Inheritance of Coat-colour in Thoroughbred Horses. — (a.) Direct Line. 
First Degree, — Having regard to the probable errors — ^about '03 — in the values of 
the correlation coefficient r^yy it seems quite reasonable to suppose that the mean 
parental correlation, actually '5216, is practically '5. It is quite impossible to 
imagine it the '3 of Mr. Galton's view of the Law of Ancestral Heredity. If we 
adopt the view of that law given in my paper on the Law of Ancestral Heredity, t 
and take the coefficient y to be different from unity, then it is shown in my paper on 
the Law of Reversion^ that we cannot on the theory of blended inheritance get 
parental correlation as high as '5 without values of the fraternal correlation which 
are much higher than those hitherto observed, certainly much higher than, as we 
shall see later, we find in the case of coat-colour in horses. Coat-colour in horses does 
not thus appear to be at all in accord with Mr. Galton's view of ancestral inheritance, 
or even with my generalised form of his theory. It does accord very well with what 
we might expect on the theory of exclusive inheritance as developed above, p. 91, 
on the assumption that there is no reversion. 

Looking at the matter from the relative standpoint, we see that not much stress can 
be laid on the respective influences of the sire and dam on the colt, or of the sire and 
dam on the filly ; but, on the other hand, the filly does appear to inherit more from 

* See the concluding remarks in the memoir on " Genetic (Reproductive) Selection," * Phil. Trans.,' 
A, vol. 192, pp. 257—330. 

t • Roy. Soc. Proc.,' vol. 62, p. 386 et seq. 
J 'Roy. Soc. Proc.,' vol. 66, p. 140 et seq. 



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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 99 

both parents than the colt does. There is certainly (judged from coat-colour) no 
preponderance of the sire's influence over the dam's such as breeders appear occasion- 
ally to imagine. The average influence of the dam on the ofispring indeed appears 
to be slightly greater than that of the sire, but the difference is of the order of the 
probable error, and not of the overwhelming character exhibited in the case of Basset 
Hounds. There is indeed in the case of thoroughbred horses not the same chance of 
carelessness produced by a misalliance afterwards screened by the defaulter. There 
exists, however, a far greater premium — considering the great value of yearlings from 
fashionable sires — set upon dishonesty. Again it is possible that when stallions 
receive too many public or private mares, or are still used in their old age, that they 
may, without losing the power of fertilising, lose some of the power of transmitting 
their character. The divergences, so far as the probable errors are concerned, are 
not such that we are forced out of our way to explain them. With the single 
exception of sire and colt we see that our table shows the universal prevalence of the 
rule that : 

Relatives of the same sex are more closely correlated than relatives of the same 
grades of the opposite sex. Thus : — 

A colt is more like his sire than his dam. 

A filly is more like her dam than her sire, 

A dam is more like her filly than her colt. 

A grandsire is more like his grand-colt than his grand-filly. 

A colt is more like his brother colt than his sister filly. 

A filly is more like her sister filly than her brother colt. 

the latter two cases being true for both whole and half siblings. 

The solitary exception is that a sire is more like his filly than his colt. 

If we were to assume it a rule that a filly in the matter of coat-colour has stronger 
inheritance all round than a colt, we should find it agree with our results for parental 
inheritance, and receive considerable support for the much stronger correlation of 
fillies than of colts, when either whole or half siblings. But it would not be in 
accordance with our results for grandparents, for which, however, we have only 
details for two out of the eight possible cases. On the whole, I think we must 
content ourselves with the statements that parental correlation is certainly about '5, 
and that with high probability each sex is more closely correlated with its own sex of 
the same grade of relationship. 

(6.) Direct Line, Second Degree. — My data here are unfortunately only for two 
cases out of. the possible eight. I hope some day to finish the series, but the labour 
of ascertaining from the studbooks the coat-colour of 700 or 800 separate sires is 
\rery great. Indeed it is not easy to foUow up the pedigree through the male line 
when the sire is not one of the famous few. On the other hand, it is much easier 
through the female line. For this reason the maternal grandsire was taken. Even 

O 2 



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100 PROFESSOR K. PEARSON AND DR, A. LEE ON 

in this case we had to seek back for each sire — the year of whose birth was unknown 
— ^until we found the record of his coat-colour given under the heading of his dam in 
the year of his birth. 

The average of our two cases gives a coefficient of correlation == '3353, the colt 
having a greater degree of resemblance to the grandsire than the filly. This value is 
substantially greater than the '25 we might expect for exclusive inheritance, and 
more than double the value '15, to be expected for the grandparental correlation 
with Mr. G Alton's unmodified law for blended inheritance. Of course the '25 is to be 
expected as the mean of the eight grandparental series, and, as we shall see for eye- 
colour in man, these may vary very much in magnitude. But allowing for this, it 
seems quite impossible that the average value could be reduced to '15. I take it 
therefore that the grandparental, like the parental, data point to a law of inheritance 
other than that which has been described in my paper on the Law of Ancestral 
Heredity. This peculiar strengthening of the grandparental heritage has already 
been noted by me in my paper on the Law of Reversion,* and the difficulties of 
dealing with it on the principle of reversion therein discussed. There may be some 
opinion among breeders as to the desirability of emphasising the dam's strain in the 
choice of a sire which leads to this result, but if so it is unknown to me, nor do I see 
how it would work without also emphasising the correlation of the dam and foaL 
The mean value of the correlation for the maternal grandfather and grandchildren for 
eye-colour in man is '3343 — a. result in capital agreement with that for coat-colour in 
horses. In that case the average of the eight series, as we shall see later, is con- 
siderably above '25, and we must, I think, suspend our judgment as to whether this 
could possibly in the case of horses be the true mean value. As to the value '15 
it seems quite out of the question. 

As already remarked, the influence of the maternal grandsire (unlike that of the 
sire) is substantially greater on the colt than on the filly. 

(c.) Collateral Heredity, Fi7'st Degree. — Here we have more ample data to go 
upon, namely, a complete set of six tables of both whole and half siblings of both 
sexes. 

We notice one or two remarkable features straight off. In the first place, in the 
case of both fillies and colts, the whole siblings of the same sex have not a correlation 
the double of that of the half siblings, but have a correlation very considerably less 
than this. A priori we might very reasonably expect the one to be the double of 
the other. This is what would happen in the case of blended inheritance on the 
hypothesis of equipotency of the parents. As the half siblings are on the dam's side, 
we might assert a considerable prepotency of the dam over the sire. This cannot 
indeed be the explanation of the divergence in the case of Basset Hounds, where the 
half siblings have a correlation considerably less than half that of whole siblings, t 

* *Roy. Soc. Proc.,' vol. 66, p. 140 et seq. 
t * Roy. Soc. Proc.,' vol. 66, p. UO dseq. 



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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 101 

and yet the prepotency of the dam in coat-colour is very marked. But in the 
present case there is on the average only a slight, if indeed it be a real, prepotency of 
the dam. Further, if we turn to the correlation, no longer of siblings of the same 
sex, but of opposite sexes, we find the correlation of the whole siblings is approxi- 
mately double that of the half siblings, as we should d priori have expected. 

Taking averages on the assumption that the correlation for whole siblings should 
be double that for half siblings, we have the following results : — 

Correlation between colts based on results for whole and half siblings . '6667 

Correlation between fillies based on results for whole and half siblings . '7729 
Correlation between filly and colt based on results for whole and half 

siblings '5747 

Mean correlation of siblings based upon all results for whole siblings . '6329 

Mean correlation of siblings based upon all results for half siblings . . '7100 
Mean correlation of siblings based upon results for both whole and 

half siblings '6714 

We can draw the following conclusions : — 

(i.) In whatever manner we deduce the fraternal correlation it is very much larger 
than the '4 for whole brethren, or the '2 for half brethren, required by the unmodified 
Galtonian law. Such values, as we see above, could be deduced fi-om the modified 
Galtonian law by taking y greater than unity,* but this would involve values for 
the parental correlation sensibly less than those given by theory. We are again 
compelled to assert that the modified as well as the unmodified theory of blended 
inheritance, based on the Galtonian law, does not fit the facts. The above values, 
however, are quite compatible with the theory of exclusive inheritance on the 
assumption that there is an individual (not a sexual) prepotency from one pairing to 
another. 

(ii.) In whatever way we consider it, it would appear that the average value of 
the fraternal correlation, as deduced fi:om siblings with the same dam only, is 
greater than that deduced from siblings with both the same dam and the same sire. 

I am not able to explain this in any way, for I cannot assert a very substantial 
prepotency of the dam. All I can say from the data at present available is that for 
horses and dogs there appears to be no simple numerical relation between the correla- 
tion of whole and half brethren. 

(iii.) Offspring of the same sex are more alike than offspring of opposite sexes. 

This appears to be generally true, so far as our data at present extend, and will be 
fairly manifest fi^om the table below. 



* 'Boy. Soc. Proc.,' vol. 66, p. 140 e^ seq. 



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102 



PROFESSOR K. PEARSON AND DR. A. LEE ON 
Table V. — Collateral Heredity. 



1 

i 

Pair. 

! 


Man. 


Dog. 


Horse. 


Stature.* 


Cephalic 
Index.! 


Eye-colour. J 


i 

Coat-colour.§ 

1 


Coat-colour. 


Whole 
Siblings. 


HaH 
Siblings. 


Brother-Brother . i 
Sister-Sister . . 
Brother-Sister . : 


•3913 
•4436 
•3754 


•3790 
•4890 
•3400 


•5169 
•4463 
•4615 


i} -5257 1 


•6232 
•6928 
•6827 


•3561 
•4265 
•2834 



It will be noted that, with the single exception of eye-colour in man, sister and 
sister are more alike than brother and brother. 

The mean value of the fraternal correlation for stature is '4034, and for cephalic 
index '4027. These results are in excellent accordance with the '4 required by the 
Galtonian theory of blended inheritance. The mean values for eye-colour in man, 
coat-colour in dogs, and coat-colour in horses are : '4749, '5257, and '6329. These 
are quite incompatible with that theory. I venture accordingly to suggest that we 
have here cases of an inheritance which does not blend, and that it is an inheritance 
which is far more closely described by the numbers we have obtained on the theory 
before developed of exclusive inheritance than by the law of ancestral heredity. 

Taking in conjunction with these results for collateral heredity, those for parental 
and grandparental inheritance, we see that coat-colour in horses diverges widely from 
the laws which have been found sufficient in the cases of stature and cephalic index 
in man. The latter characters are really based on a complex system of parts, while 
the determination of coat-colour may depend on a simple or even single factor in the 
plasmic mechanism. Here Mr. Galton's suggestion of an exclusive inheritance of 
separate parts ('Natural Inheritance,' p. 139) may enable us to understand why 
stature and cephalic index differ so widely in their laws of inheritance from coat- and 
eye-colours. 

Part HI. — On the Inheritance of Eye-Colour in Man. 

(10.) 071 the Extraction and Reduction of the Data. — The eye-colour data used in 
this memoir were most generously placed at my disposal by Mr. Francis Galton. 
They are contained in a manuscript wherein, by a simple notation, we can see at a 

* Pearson, * Phil. Trans.,' A, vol. 187, p. 253 et seq. See Note I. at the end of this paper. 

t Pawckti and Pearson, * Roy. Soc. Proc.,' vol. 62, p. 413 et seq, 

X Present memoir, p. IIS et seq, 

§ Pearson, *Roy. Soc. Proc.,' vol 66, p. 140 et seq. 



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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 103 

glance the distribution in eye-colour of a whole family in its numerous male and 
female lines. Such complete details of the various direct and collateral relationships 
I have not hitherto come across, and from them I was able to form, in the course of 
some months of work, the twenty-four tables of correlation which are given in 
Appendix II. These tables, for the first time, give the whole eight series of grand - 
parental and the whole eight series of avuncular relationships, besides such as we have 
deduced for other characters previously, i.e., the four parental, the three fraternal rela- 
tionships, and the table for assortative mating. The very great importance of this 
material will at once be obvious, and I cannot sufficiently express my gratitude to 
Mr. Galton for allowing me to make free use of his valuable data. 

At the same time we must pay due regard to the limitations of this material, which 
it is needful to enumerate, so that too great stress may not be laid on the irregularities 
and divergences which arise when we attempt to reduce the results to laws. These 
limitations are as follows : — 

(a.) While the data of about 780 marriages are given in the record, they belong 
to less than 150 separate families. All our relationships, therefore, contain pairs 
weighted with the fertility of the individual families. Thus it was necessary to enter 
every child of a mother, every nephew of an uncle, and so forth. In the horse data 
we could take 1000 distinct mares and give to each one foal only. That is not possible 
in the present case. 

(6.) The colour of eyes alters considerably with age. It is not clear that some of 
the eye-colours are not given for infants under twelve months, and certainly the eye- 
colours in the case of grandparents and others must have been taken in old, perhaps 
extreme old, age. In a large number of other cases of great grandfather, great great- 
grandfather, &c., great uncles, and so forth, the eye-colours must have been given 
from memory or taken from portraits — in neither alternative very trustworthy sources. 
Thus while the horse colour is always given for the yearling foal by the breeder, the 
eye-colour is given at very different ages, and comes through a variety of channels. 

(c.) The pqysonal equation in the statement of eye-colour, when the scale contains 
only a list of tint-names is, I think, very considerable. The issue for the collection of 
data of a plate of eye-colours like that of Bertrand would be helpful, but we can 
hardly hope for a collection of family eye-colours so comprehensive as Mr. Galton's 
to be again made for a long time to come. 

These causes seem to me to account for a good deal of the irregularity which appears 
in the numerical reduction of the results, but they are not, I hold, sufficient to largely 
impair the very great value of Mr. Galton's material. 

In tabulating the data, I have followed the scale of tints adopted by Mr. Galton, 
and I have used the entire material available in the cases of the grandparental, 
avuncular, and marital relations. I nearly exhausted the data for the parental 
relationships, but in these tables, which were first prepared, I stopped short at 1000 
for the sake of whole numbers. I found, however, that it did not make the arithmetic 



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104 PROFESSOR K. PEARSON AND DR. A. LEE ON 

sensibly shorter, and I afterwards dropped this limitation. In the case of brethren 
I took 1500 of each case — I daresay I could have got 2000 out of the records. As 
the light-eyed brethren are entered ^r^^ in Mr. Galton's MS., the First Brother 
in my unsymraetrical tables is always lighter-eyed than the Second Brother, hence 
the tables had to be rendered symmetrical by interchanging and adding rows and 
columns before we could reduce them. Thus the symmetrical tables have an apparent 
entry of 3000 pairs. Of course 1500 is the number used in finding the probable 
error of the correlation coeflficient. The like difficulty does not occur in the brother- 
sister table, where indeed the difference of mean eye -colour for the two sexes would 
not allow of our making the table symmetrical. A comparison of the symmetrical 
with unsymmetrical tables for colts-colts and fillies-fillies, will show how little need 
there is for rendering the tables symmetrical when pairs are taken out of any similar 
class and tabulated without regard to the relative magnitude of the character in the 
two individuals of the pair, i.e., Weatherby's record places the individuals simply in 
order of birth and not of darkness or lightness of coat-colour. 

Table VII. gives the value of the chief numerical constants deduced from the twenty- 
four eye-colour tables in Appendix II.* 

(11.) On the Mean Eye-colour having regard to Sex and Generation. — In order to 
test the degree of weight to be given to our conclusions, I have drawn up a table o* 
probable errors for four typical cases — cases by no means selected to give the 
lowest possible values. Further, in Table VIII. I have given the probable error in 
the position of the median as determined in terms of the grey, blue-green range by 
the modification of Mr. Sheppard's formula (see p. 95). The grey, blue-green range 
of eye-colour is about one-fifth of the total observed range, so that the probable error 
in the position of the median varies from about "4 to 1 per cent, of that range. This 
is not a large error, but, relative to the small variations of value with generation and 
.sex, it is sensible, and we must not draw too fine conclusions on the basis of single 
inequalities. 



Table VI. Table of Probable Errors 


in Eye-colour Data. 


Relations. 


Vx 


Vy 1 f '* ; ^*' ^^ 


1 
Mother and Son . . -0253 
Maternal Grandmother 

and Granddaughter .! -0348 
Sister and Sister . . 
Maternal Aunt and -0244 

Nephew -0230 


•0188 

•0350 

•0244 
•0186 


•0431 
•0767 

•0414 


•0267 

•0276 

•0216 
•0255 


•0256 -0283 

•0314 -0361 

•0216 -0234 
•0250 ^0302 

1 



* The theoretical formulae by aid of which these constants were determined, have been indicated in the 
earlier part of this memoir, and in Part VII. of the present series on Evolution. The actual work of 
reduction has been most laborious, but I trust that our results are free from serious error. 



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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 105 

If we examine this table we see that the error in rj amounts to fix)m '02 to '025 
when we have upwards of 1000 tabulated eases, but can amount to '035 when we 
have as few as 700 to 750 tabulated cases. An examination of the values of rj in 
Table VII. shows us that most of our differences with probable errors taken on this 
scale are very sensible. A comparison with Table VIII. shows us that the probable 
error of the median is always greater than the probable error of 17, and accordingly 
the former, being much easier of calculation, may be taken as a maximiun limit. The 
probable errors of f , t.c., the ratio of cTar to o-y, are more considerable, amounting to about 
•04 for our longer series, and even to "077 in the case of the short series of grand- 
mother and granddaughter, but in this case f actually takes its maximiun value 
of 1'291, so that the error is under 6 per cent. ; in the longer series it is under 5 per 
cent. Again, we see that in most cases our differences in the ratio of variabilities are 
quite sensible. It must be admitted, however, that the ratio of variabilities as based 
on the grey blue-green range of eye-colour is not as satisfactory as that based on the 
bay range of coat-colour in horses. In the latter case, one-half of the horses fall into 
the bay range, but only about a quarter of mankind fall into the grey blue-green 
range of eye-colour, and, fiirther, the appreciation of eye-colour seems to me by no 
means so satisfactory as that of coat-colour in horses. 

I have tried a further series of values for the ratio of the variabilities by measuring 
the ranges occupied not only by the tints grey blue-green, but by the whole range of 
tints 3, 4, 5, and 6 of Mr. Galton's classification (see p. 67). Lastly, I have taken 
a third method of appreciating the relative variabilities, namely, by using the method 
of column and row excesses, E^ and E^, discussed in Part VII. of this series. While 
this method has the advantage of using all and not part of the observations to deter- 
mine the ratio of crx to o-y, and so naturally agrees better with the results based on 
the four than the one tint ranges, it suffers from the evil that these excesses can only 
be found by interpolation methods, which are not very satisfactory when our classes 
are, as in this case, so few and so disproportionate. On the whole, this investigation 
of relative variability is the least satisfactory part of our eye-colour inquiry, and I 
attribute this to two sources : — 

(i.) The vagueness in appreciation of eye-colour when no colour scale accompanies 
the directions for observation {cf. p. 103, (c) ). 

(ii.) A possible deviation from true normality in the factor upon which eye-colour 
really depends {cf. p. 80, (ii) 80). 

Lastly, we may note that the probable error in the correlation amounts in most 
cases to less than '03, rising only somewhat above this value for grandparental 
inheritance, where our series are somewhat short — 650 to 750 instead of 1000. Here 
again most of the divergences are quite sensible. 

Allowing accordingly for the comparative largeness of our probable errors, we 
shall do best to base conclusions on the general average of series ; to insist on general 
inequalities rather than on exact (juantitative differences, and to eni4)lia8i8e the 

VOL. CXCV. — A. P 



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106 



PKOFESSOR K. PEAESON AND DR. A. LEE ON 



I 
i 



I 

I 
? 



Num- 
ber 
of 

cases. 


^ 


1000 
1000 
1000 
1000 


1500 
1500 
1500 




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Coefficients of 
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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 



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108 PROFESSOR K. PEARSON AND DR. A. LEE ON 

general tendency of a series rather than pick out single differences for special 
consideration. If we do this we shall still find that remarkable results flow 
from our Tables VII. and VIII., most of which seem hitherto to have escaped 
attention. 

I return now to the special topic of the present section, the mean eye-colour, after this 
lengthy — if needful — digression on the probable error of the data given in our tables. 

We may, I think, safely draw the following conclusions : — 

(a.) Man has a mean eye-colour very substantially lighter than that of woman. 

If we compare the mean eye-colour of father with mother, of son with daughter, 
of brother with sister, of grandfather with mother, of uncle with aimt, of grandson 
with granddaughter, of nephew with niece, we have the same result — man is distinctly 
lighter eyed than woman. 

(6.) Tlbcre appears to be a secular change taking place in eye-colour^ but this 15 
more marked and definite in the man than in the woman. 

Thus we have the following mean values for 77 in classes, which must roughly 
represent successive generations : — 

Grandfather. . . -3658 1 .^aac. Grandmother . . '8757 1 .ggoo 



3658 -1 .4,49 
5241 J 


Grandmother . 
Mother . . . 


. -8757 
•8290 


5^29 \ -6484 
7039 J 


Daughter. . . 
Granddaughter. 


. -7524 
•8508 



}^ 



Father 

}• 



^"^ •^•'"•' )^ -6484 -— 8— — . . . '—J. .8016 

Grandson. . . , 

Another comparison may be made by noting that grandsons are darker than 
grandfathers, sons than fathers, nephews than uncles. Similarly, granddaughters are 
lighter than grandmothers, daughters than mothers, but nieces are not lighter than 
aunts, as we might have expected. Thus, while the records show a definite darkening 
of the eyes of men, there appears to be a certain but less sensible lightening of the 
eyes of women. Again, the younger generations are much closer in eye-colour than 
the older generations. The difierence in eye-colour between grandsons and grand- 
daughters, sons and daughters, nephews and nieces is only about 15 per cent, of the 
grey blue-green range, but for fathers and mothers it is 30 per cent., and for grand- 
fathers and grandmothers 50 per cent. 

When we realise that difference in eye-colour appears to be a sexual character, the 
true explanation of this secular change in eye-colour becomes stiU more obscure. 

If the lighter eye-colour of the older generation be due to an effect of old age, why 
is it conspicuous only in the male and not in the female ? Why is the mother sensibly 
darker than the daughter, but the father sensibly lighter than the son ? 

Further, supposing light eyes much commoner among our grandfathers than among 
their grandsons, and dark eyes among our grandmothers than among their grand- 
daughters, we cannot attribute the great approach in eye-colour to a blending of the 
parental characters, for, as we shall see later, eye-colour does not seem to blend, it is 
rather an exclusive character. We should, therefore, be thrown back on prepotency 



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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 109 

of the mother — a conclusion possibly warranted by our results in the case of 
daughters, but not in the case of sons. Again, why was there such a marked 
difference in eye-colour between the men and women of three or four generations 
back ?* And if it was a sexual character, why is it disappearing ? Was it not, 
perhaps, a racial difference ? Light and dark eyes are not unusually associated with 
distinct races, and it is just possible that the change in eye-colour is a product of 
reproductive selection ; the old blue-eyed element of the population may be dwindling 
owing to the greater fertility of the women of dark-eyed race, and thus without any 
obvious struggle for existence and survival of the fitter, the blue-eyed race may be 
disappearing from England, as the Langobard type has so largely gone from Italy and 
the Frank from France, t It will not do to be dogmatic about these matters, but the 
more one measures characters in different generations, the less stable do races appear 
to be. We speak of the national characters of the Englishman or the Frenchman 
based upon our experience of how these races have acted in past history, but 
although there has been no great racial invasion nor struggle, can we safely assert 
the physical characters of the Englishman to-day do not differ substantially from 
those of the Englishman of the Commonwealth ? It seems to me that the possibly 
continuous change of characters in a race, not subjected to very apparent internal or 
external struggle, is a problem of the highest interest to the anthropologist and 
.ultimately to the statesman. 

Whatever be the explanation of this secular change in eye-colour, it appears to 
correspond singularly^ enough to the secular change we have noted in the coat-colour 
of thoroughbred horses — in the older generation the sexes differ more widely than in 
the yoimger. 

(e.) TJie maternal male relative {grandfather^ and uncle) is substantially lighter-eyed 
than the patei^nal male relative {grandfather and uncle). — I see no explanation for this 
curious result, but it seems worth while to specially note it, for there are curious 
anomalies in the inheritance through the various male and female lines which may 
find their complete explanation some day when more and possibly more trustworthy 
characters have been investigated. 

(12.) On the Variability of Eye-colour with reference to Sex and Class. — The 
determination of the relative variability of not exactly measurable characters is, as 
we have already seen (p. 105), a somewhat delicate problem. It is more so in the case 
of eye-colour in man than of coat-colour in horses, for there is greater difference in 
the means, and accordingly the ratio of crx/o-y, as found from the ratio of the 
" excesses" (p. 105), will be even less reliable.^ The class indices corresponding to the 

* Mr. JJalton's records went back to great-grandfathers, many of whom accordingly appear in our 
data for grandfathers. 

t See Note II. at the end of this paper. 

J The relative variability of all classes was worked out for thorough-bred horses by the " excess " 
method, and in only one case — that of dam and colt — did it differ from the bay range method in its 
determination of the class with the greater variability. 



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110 PROFESSOK K. PEARSON AND DR. A. LEE ON 

grey blue-green range are also not entirely satisfactory in their results, nor those 
taken for a still larger range covering tints 3, 4, 5, and 6, or blue-green, grey, hazel, 
light brown, and brown, which cover roughly about 1*5 to 1'6 times the standard 
deviation. We shall now consider the results of three methods of considering the 
relative variability, (a) jfrom the excesses given in columns 1 and 2 of Table VIII. ; 
(fi) from the grey blue -green range given in columns 3 and 4 of Table VIII. ; and (y) 
from the range of tints 3 to 6 inclusive given in columns 5 and 6 of Table VIII. As 
we have already indicated, these methods are not likely to give the same relative 
magnitude numerically for the variabilities ; we must content ourselves if they agree 
in making the ratio of cr^ to Cy greater or less than unity. Now, in the twenty-two 
cases 

a and fi disagree in 10 cases. 

15 and y disagree in 7 cases. 

a and y disagree in 5 cases. 

Further, for the five cases in which a and y disagree, those for father and son, 
paternal grandfather and grandson, maternal uncle and nephew, show so little 
difference of variability in the two sexes that both methods give sensibly the same 
results, i.e., equality of variability. In the cases of the paternal grandfather and 
grandchildren, the two methods diverge rather markedly. 

It will be of interest accordingly to work out the probable errors as given by the 
excess method for one, say the first of these cases. The theory is given in Part VIL of 
the present series. Here Ei = 275*165, E^ = 309"013, whence we find probable error 
of El = 17*273, probable error of Eg = 16*925, correlation between errors in E^ and 
Eg = — -4424, probable error in crj/crg = '0394. 

Thus the probable error in the ratio of the variabilities is about 4 per cent., and of 
the order of the quantities by which we are distinguishing the relative size of Ci 
and (Tg. 

Further, there is another source of error in evaluating Ei and Eg due to the 
method of interpolation used, and this would still further increase the probable error 
in cTi/cTg. We cannot therefore lay any very great stress on the manner in which the 
ratios of variabilities for the paternal grandfather and grandchildren have swung 
round from (a) to (y). 

A further examination shows us that in all five cases wherein y differs from a it is 
in accord with fi. I shall accordingly take y as the standard criterion, but in those 
cases in which it has agreement with a, its conclusions must be given greater weight. 

(a.) On the Relative Variability of Sex in Eye-colour. — The following male groups 
are more variable than the corresponding female groups : — 

Sons of fathers than daughters of fathers. 
Sons of mothers than daughters of mothers. 
Brothers of brothers than sisters of sisters. 



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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. Ill 

Grandsons (in four series) than granddaughters (in four same series). 
Nephews (in four series) than nieces (in four same series). 
Fathers (in two series) than mothers (in two series). 
Grandfathers (in foiu* series) than grandmothers (in four like series). 
Uncles (in four series) than aunts (in four like series). 

The following female groups are more variable than the corresponding male 
groups : — 

Sisters of brothers than brothers of sisters. 
Wives than husbands. 

We have thus 21 series with male preponderance against only two with female 
preponderance of variability. 

Again, the mean range of tints 3, 4, 5, 6 in 22 male series equals 1*5424 cr^, and 
in 22 female series equals 1*6740 o-y, or we have enough evidence to show that the 
ratio of male to female variability is about 1*08.* 

This greater variability of the male in eye-colour is of considerable interest. It 
does not appear to be a result of sexual selection, for so far as our comparatively small 
series weighs, husbands are less variable than wives. That mothers are, however, less 
variable than fathers seems to indicate that dark-eyed women are more fertilet than 
light-eyed, for we must bear in mind that mothers have on the average a darker eye- 
colour than wives. We have thus again reached the same conclusion as before, 
namely, that a dark-eyed element in the population with a prepotent fertility is 
replacing the blue-eyed element. 

The other female exception to the general rule of greater variability in the eye- 
colour of the male is that in mixed families the sisters appear to be more variable 
than their brothers, notwithstanding that brothers of brothers are more variable than 
sisters of sisters. In other words, so far as eye-coloiu* is concerned an exceptional 
man is more likely to have brothers than sisters, but an exceptional woman also is 
more likely to have brothers than sisters. The inference is not very strong, as the 
excess method (a) makes brothers of sisters and sisters of brothers of sensibly 
equal variability; it rests therefore on (fi) and (y) only. Still it deserves ftdler 
investigation. 

(6.) Let A and B be two grades of relationship, of which A refers to the older 
generation, and A and B refer to either sex. Then the variability of all the A's 

* It is worth noting that the ratio of male to female variability in the coat-colour of horses is r05 (see 
p. 96). In both cases the female is darker, i.e., has less of " colour " ; thus if we could take a coefficient of 
variation ratio instead of standard deviation ratio as the test, we should find the difference of variability 
less, possibly even zero. 

t For if mothers are to ]ye less variable than wives, their distribution must be more compressed round 
the mean than that of wives ; this denotes that fertility is correlated with eye-colour, and the darker eye- 
colour goes with the greatei fertility. [See Note II. at end of memoir, however.] 



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112 PROFESSOR K, PEARSON AND DR. A. LEE ON 

who have female B's is invariably greater than the variability of all the A's who 
have male B's. 

The law appears to be universal, at least it is absolutely true for all the 10 cases to 
which we can apply it. Thus the father of sons is less variable than the fether of 
daughters, the maternal grandmother of grandsons less variable than the maternal 
grandmother of granddaughters, or the paternal uncle of nephews less variable than 
the paternal uncle of nieces. In other words, although women appear, in eye-colour, 
to be less variable than men, they spring from more variable stocks. 

This law is a remarkable one, but in face of the evidence for it, it seems difficult to 
doubt its validity. Should it be true for more characters in man than eye-colour,* 
the conclusions to be drawn from it will be somewhat far-reaching, however difficult 
it may be to interpret its physiological significance. 

(c.) On the Relative Variability of Different Generations. — We have already had 
occasion to refer to the general rule that the older generation will be found less 
variable than the younger, for it is in itself a selection, namely, of those able to 
survive and reproduce themselves. But this rule is obscured in the present case by 
several extraneoxis factors, thus : — 

(i.) The male is sensibly more variable than the female, consequently it is quite 
possible that an elder male generation should appear more variable than a younger 
female generation. 

(ii.) There appears to be a secular change in eye-colour going on. Thus while the 
grandparental population is a selection from the general population, the general 
population, at a given time, is a selection from that of an earlier period. 

Thus, taking means in the cases of the grandparental and avuncular relationships, 
we have from (y) the following results : — 

The father is more variable than son and than daughter. 

The mother is less variable than son and more than daughter. 

The grandfather is more variable than grandson and than granddaughter. 

The grandmother is less variable than grandson and more than granddaughter. 

The uncle is more variable than nephew and more than niece. 

The aunt is less variable than nephew and more than niece. 

In other words, the older generation is always more variable than the yoimger, 
except when rule (a), that the male is more variable than the female, comes in to 
overturn this law. If we confine ourselves to comparisons of the same sex the rule is 
seen to be universal. 

We are thus forced again to ask for an explanation of the decreasing variability of 
eye- colour, and can only seek it in that secular change we have several times had 

* Fathers of daughters are more variable in stature than fathers of sons (*Phil. Trans.,' A, vol. 187, 
p. 274). I propose to reinvestigate the question with regard to mothers from the material of my family 
measurement cards, which is far more extensive than the material I had at my disposal in 1895. 



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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 113 

occasion to refer to. Mean and standard deviation of eye-colour appear to have 
changed sensibly during the few generations covered by Mr. Galton's eye data. 

It is difficult to understand how any obsciu^ity about the recording of eye-colours 
could lead to anything but chaos in the numerical results. It does not seem to 
me possible that such results as we have reached under (a), (6), and (c), namely, 
greater variability in the male, greater variability in the stock of the female, and 
secular change in variability, can be due to any process of recording. I am forced to 
the conclusion that they are peculiar to the character under investigation, and are 
not due to the manner of taking the record or of dealing with it arithmetically. I 
have purposely avoided drawing attention to small diflPerences and forming any con- 
clusions which did not depend on whole series of groups and substantial averages. 

1(13.) On the Inheritance of Eye-colour, (a.) Assortative Mating. — Before we 
enter on the problem of inheritance, it is as well to look at the substantial 
correlation obtained between the eye-colour in husband and wife. When in 1895 
I reached the value '0931 ± '0473 for stature, I wrote, " we are justified in con- 
sidering that there is a definite amount of assortative mating with regard to height 
going on in the middle classes."* Since then we have worked out the coefficients of 
correlation in stature, forearm, and span for 1000 husbands and wives (instead of 
200) fi-om my family datat cards. The results, which are very substantial, will be 
dealt with in another paper, and amply confirm my view that assortative mating is 
very real in the case of mankind. The value ('0931) cited above is in close agree- 
ment with the result now reached ('1002 db '0378) for eye-colour in the same 
materiaL The correlation between husband and wife for two very divergent 
characters is thus shown to be about "1, or is 25 per cent, greater than is required 
between first c(msins\ by the law of ancestral heredity. 

This remarkable degree of likeness between husband and wife — ^the scientific 
demonstration that like seeks like — cannot be overlooked. It shows that sexual 
selection, at least as far as assortative mating is concerned, is a real factor 
of evolution, and that we must follow Darwin rather than Wallace in this 
matter. § * 

(6.) Collateral Heredity. First Degree. — I deal first with this form of heredity, 
as it offers least points for discussion. The values of the correlation '5169 for 
brothers, and '4463 for sisters and sisters are considerably less than what we have 
found for coat-colour in horses, but, like the value '4615 for brothers and sisters, are 
substantially greater than '4 to be expected from the immodified Galtonian law. 
They could be reached by making y greater than unity in my statement of the law 
of ancestral heredity. || They could also be given by the law of exclusive inheritance 

* * Phil. Trans.,' A, vol. 187, p. 273. 

t See also * Grammar of Science,' second edition, pp. 429-437. 

X ' Roy. Soc. Proc.,' vol. 62, p. 410. § « Roy. Soc. Proc.,' vol. 66, p. 140 ei seqS 

II I have considered possible explanations of thia apparently large assortative mating (i.) in both stature 
VOL. CXCV, — A. 9 



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114 PROFESSOR K. PEARSON AND DR. A. LEE ON 

(see p. 90) with a certain degree of prepotency in the individual pairing. As we 
have already noted, collateral inheritance of the first degree alone considered 
will not enable us to discriminate between blended and exclusive inheritance. 

We note that the male in collateral inheritance predominates over the female, 
brothers being more alike than sisters in eye-colour, and brother and sister more 
alike than sister and sister. The mean value for inheritance in the same sex is, 
however, greater than the value for inheritance between opposite sexes {cf, p. 102). 

(c.) Collateral Heredity. Second Degree, — A very cursory inspection of the 
coefiicients of correlation for the eight series of avuncular relationships shows us 
that it is quite impossible that the mean value should be '15 as required by the 
Galtonian Law. The average value of the avuncular correlation is '2650, and of 
the regression of nephew and niece on uncle or aunt is '2733. The probable error 
of the former result will not be more than '02, and of the latter something greater, 
as the ratio of the variabilities is open to larger error. This mean value is accord- 
ingly, within the limits of errors of investigation, identical with the '25 to be expected 
on the theory of exclusive inheritance. It is a value which appears to be quite 
impossible on the theory of blended inheritance even with my generalised form of 
the ancestral law. 

We may draw several other important conclusions from our table of avuncular 
correlations :— 

(i.) Nephews are more closely related to both uncles and aunts than nieces are. 
This is true in each individual case, whether it be judged by correlation or regression. 
The mean correlations for uncles and for aunts are as '3081 to '2219 respectively. 

(ii.) Uncles are more closely related to nephews and nieces than aunts are. This 
is true for three out of the four individual cases ; in the fourth case the difference 
is of the order of the probable error of the difference. The mean correlations ot 
nephews and nieces are as '2923 and "2377 respectively. 

(iii.) Paternal uncles and aunts are more closely correlated with their nieces and 
nephews than maternal uncles and aunts. The mean values are as '2719 to '2580. 

(iv.) Resemblance between individuals of the same sex is closer than between 
individuals of opposite sex. The mean values for the avuncular correlation in the 
same sex and in the opposite sex are respectively "2751 and •2549. 

(v.) Uncles are more closely related to nephews than aunts to nieces (mean 
correlations as '3455 to '2046). In fact, generally, we see a very considerable 
preponderance of heredity in the male line so far as these avuncular relations for 

and eye-colour, being characters of local races, or even families, and the husband seeking his wife in his 
own locality or kin; (ii.) in a possible coiTelation of homogamy and fertility. See *Eoy. Soc. Proc,,* 
vol. 66, p. 28. Neither seem very satisfactory. Consciously or unconsciously, man and woman appear to 
select their own type in eye-colour and stature, until they are apparently more alike than such close blood 
relations as first cousins ! Until we know how far this correlation extends to other characters, it would, 
perhaps, be idle tp draw conclusions as to its bearing on widely current views as to first cousin marriage». 



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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 115 

eye-colour extend. It is noteworthy that while the two highest correlations are 
reached for nephew with paternal and with maternal uncles, nearly the two lowest 
are found for niece with paternal and with maternal aimts. Without laying 
special stress on each small diflTerence, it must be admitted that the avuncular 
correlations vary in a remarkable manner with sex, and differ very widely from 
the practical equality of resemblance which we might d prioii have expected to 
exist in this relationship. 

(d.) Direct Heredity. First Degree. — Here we have a mean value of the paternal 
correlation = '4947. This is in excellent agreement with the '5 to be expected by 
our theory of exclusive inheritance ; it is thus in practical agreement with the value 
of the parental correlation obtained for the inheritance of coat-colour in horses. It 
would not be inconsistent with a high value for y in the theory of blended inheri- 
tance, but such a value of y is rendered impossible by the values we have obtained 
for collateral heredity (see 'Roy. Soc. Proc.,' vol. 66, p. 140 et seq.). 

We may draw the following special conclusions: — (i.) The son inherits more 
strongly from his parents than the daughter, the mean correlations are as '5160 
to "4733 ; (ii.) The son inherits more strongly from his father than his mother, 
and the daughter more strongly from her mother than her father. 

This is part of the general principle which we have seen to hold, namely : that 
change of sex weakens the intensity of heredity. 

The correlation of father and daughter appears to be abnormally below the other 
three, but something of the same kind has been noted in certain stature data ; as it 
is, the high correlation of father and son renders the mean paternal correlation with 
o&pring ('4936) sensibly equal to the mean maternal correlation ('4956). 

(e.) Direct Heredity. Second Degree. — If we take the mean value of the eight 
grandparental correlations, we find it equals '3164, while the meaji value of the 
regression of offepring on their grandparents is '3136. These results are absolutely 
incompatible with the '15 required by Mr. Galton's unmodified theory, and they in 
fact put the theory of blended inheritance entirely out of court. At the same time, 
unlike the cases of parental, avuncular, and fraternal inheritance, they cannot be said 
to be in good agreement with the value '25 required by the theory of exclusive 
inheritance. We have to admit that our grandparental data are shorter series than 
in the other cases, and that guesses as to grandparents' eye-colour, based on memory, 
miniatures, &c., were more likely to be made. Further, such guesses might easily 
be biased by a knowledge of the eye-colour of more recent members of the family. 
Still a reduction from '32 to '25 is a very large reduction, and we have to remember 
that for long series in the case of the thoroughbred horses, with no such guessing at 
colour as may occur with ancestors' eyes, we found '3353 for the maternal grand- 
sires, a result in excellent agreement with the '3343 found for the maternal grand- 
fathers in the present case. Thus while the theory of exclusive inheritance without 
reversion suffices to describe the quantitative values we have found for the parental, 

g 2 



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116 PROFESSOR K. PEARSON AND DR. A. LEE ON 

the avuncular and the fraternal correlation in the cases of both horse and man, it is 
yet in both these cases unsatisfactory so far as the grandparental inheritance is con- 
cerned. It may be imagined that if we allowed for reversion, we might emphasise 
the grandparental correlation beyond the value '25 suggested by theory. But I 
have shown in my memoir on the " Law of Reversion," that with the parental correla- 
tion as high as '5, we cannot hope to have the grandparental correlation even with 
reversion higher than •25. (See * Roy. Soc. Proc.,' vol. 66, p. 140 et seq.) Clearly the 
values obtained for grandparental correlation in this paper — the first I believe 
hitherto investigated — seem to present anomalies which oiu* theory of blended 
inheritance totally fails to accoimt for, and which may require some modification of 
our views on reversion before we can meet them on our theory of exclusive 
inheritance. 

I note the following general results deduced from our values of the grandparental 
correlations : — 

(i.) Grandsons are more closely correlated with both grandparents than grand- 
daughters are. This is true for three out of the four cases ; the exception, maternal 
grandmother, is covered by another rule (iv.). The mean correlation for grandparents 
and grandsons is '3294, and for grandparents and granddaughters '3039. 

(iL) Grandfathers are more closely correlated with grandchildren than grand- 
mothers are. This is true in three out of the four cases, the fourth being again 
subject to rule (iv.). The mean correlations for grandfathers and grandmothers are 
'3675 and "2658 respectively. 

(iii.) Paternal grandparents appear to be more closely correlated with their 
grandchildren than maternal grandparents, the average values of the two correlations 
being '3236 and '3097 respectively. 

(iv.) Resemblance between individuals of the same sex is closer than between 
individuals of the opposite sex. The mean values for the grandparental and 
grandchild correlation in the same and the opposite sexes are '3329 and '3004 
respectively. 

(v.) Grandfathers are more closely related to grandsons than grandmothers to 
granddaughters, the mean correlations being as '3965 and '2693 respectively! It 
will be noted at once that these five rules are identical with those we have obtained 
for the avuncular correlations. So that there is small doubt that they are general 
rules relating to all grades of relationship for this character. 

It seems to me probable that the correct form of (iii.) is : Paternal grandfathers are 
more highly correlated with grandchildren ('4006) than maternal grandfathers ('3343), 
and paternal grandmothers (*2468)less highly correlated than maternal grandmothers 
(•2851). I have not stated the rule in this form, because it is not confirmed by 
the corresponding results for uncles and aunts. Paternal uncles ('3024) are more 
closely correlated with nephews and nieces than maternal uncles (*2722), but paternal 
aunts ('2414) are slightly more instead of less correlated with nephews and nieces 



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MATHEMATICAL CONTEIBUTIONS TO THE THEORY OF EVOLUTION. 117 

than maternal aunts ('2338). I consider, however, that the correlation of paternal 
aunt and nephew ('2837) in our series is abnormally high. 

Now it will, I believe, be seen that the investigation of the eight avuncular and 
the eight grandparental relationships, here made for the first time,* enables us to 
draw far wider conclusions than when, as hitherto, only parental and fraternal corre- 
lations are dealt with. In making the subjoined general statements, however, I 
must emphasise the following limitations : — 

(a.) The rules are deduced only from data for one character in one type of life. 

(fi.) This character appears to be undergoing a secular change, a change very 
possibly due to a correlation between eye-colour and fertility in wcwnaan. Thus such 
a change might not unlikely differentiate the male and female influences in heredity. 

My conclusions, definitely true for eye-colour in man, and at the very least 
suggestive for investigations on other characters in other types of life, are : — 

(L) That the younger generation takes, as a whole, more after its male than its 
female ascendants and collaterals. 

(ii.) That the younger generation is more highly correlated with an ascendant or 
collateral of the same than of the opposite sex. 

(iii.) That the younger generation is more highly correlated with an ascendant or 
higher collateral reached by a line passing through one sex only than if the line 
changes sex. 

Thus correlation is greater with a paternal uncle than with a maternal uncle, or 
with a maternal grandmother than a paternal grandmother. 

(iv.) Males are more highly correlated with their ascendants and collaterals than 
females are. 

The above rules apply to the averages ; individual exceptions will be generally 
found to arise fi:om a conflict of rules. Thus (ii.) and (iii.) may in special cases come 
into conflict with (i.). When we have more data for a greater variety of characters, 
we shall see better the relative weight of these rules in cases where they conflict. 

[f.) Exclusive InheHtance in Eye-Colour. — A cursory examination of the eye- 
colour records shows at once how rare is a blend of the parental tints. Even when 
such is recorded, it is by no means clear that we have not to deal with a medium tint 
which is really a case of reversion to a medium tinted ancestor. The failure of eye- 
colour to blend is, I think, well illustrated by what Mr. Galton has termed cases of 
" particulate " inheritance. In the thousands of eye-colours I have been through, I 
noticed some half-dozen cases only in which the two eyes of the same individual were 
of different tint, or the iris of one pupil had streaks of a second tint upon it.t 

* I anticipate equally valuable results when characters are first correlated for the nine possible cousin 
series. 

t In the same manner the occurrence of particulate inheritance in coat-colour in horses may be really 
an argument against the existence of blends. In the many volumes of the studbooks I have examined, 
the recorded instances of pieljalds are vanishingly few in niunber. 



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118 PROFESSOR K. PEARSON AND DR A. LEE ON 

If we allow that it is from the theory of exclusive inheritance that we must seek 
results in the present cases, we see that for parental, collateral, and avuncular relation- 
ships we get quite excellent results, but that the grandparental relationship is some- 
what anomalous. A priori it might appear that reversion would aid us in increasing 
the correlation between offspring and remote ascendants. But, as I have shown else- 
where,* this superficial view of reversion forgets that the parents as well as the 
offspring revert, and if we increase the grandparental correlation above '25, we at once 
reach difficulties in the values of the parental correlation, provided we adopt what 
appear to be reasonable assumptions as to reversion being a continuous and decreasing 
factor from stage to stage of ancestry. I am inclined accordingly to suspend judg- 
ment on the grandparental relationships, thinking that the smallness of the number 
of families dealt with in Mr. Galton's data (200) may have something to do with my 
peculiar results. Meanwhile I shall endeavour to get the remaining six grandparental 
tables for thoroughbred horses worked out, and see whether they confirm the high 
values ah'eady found for the two maternal grandsires and oflfepring, or give an average 
value much nearer '25. 

That the reader may see at a glance the general results hitherto obtained in this 
and other papers, I append the following table of inheritance : — 



* See my paper on "The Law of Reversion/' 'Roy. Soc. Proc./ vol. 66, p. 140 et scq. Also *The 
Grammar of Science/ second edition, 1900, pp. 486-96, " On Exclusive Inheritance." 



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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 119 

Table IX. — Theoretical and Actual Results for Inheritance. 



Relationship. 


Theory. 


Man. 


Horse. 


Hound. 


Daphnia. 


Blended 

inherit- 

ance.' 


Exclusive 
inherit- 
ance.* 


Statiu^e*. 


Head 
index.* 


Eye- 
colour.* 


Goat- 
colour.^ 


Coat- 
colour." 


Spine." 


Parental .... 
Mid-parental . . . 
Grandparental . . 
G. Grandparental . 
Avuncular . . . 
Whole sibling . . 
Half sibling . . . 


•3000 
•4242 
•1500 
•0750 
•1500 
•4000 
•2000 


•5000 

•2500 

•1250 

•2500 

•4 to 10 

•2 to -5 


•3355 

•4745 

•4034 


•3348 
•4735 

•4025 


•4947 

•3166 

•2650 
•4749 


•5216 
•3353 

•6329 
•3550 


•3507 

•1340 
•0404 

•5170 
•1646 


[•3295] 
•4660 
[•1360] 

•6934 



* Mr. Galton's unmodified hypothesis. See " Law of Ancestral Heredity," * Roy. Soc. Proc./ vol. 62 
p. 397. 

2 Without any reversion. See "Law of Reversion," * Roy. Soc. Proc.,' vol. 66, p. 140 et seq. The values 
for the fraternal correlation depend on the degree of prepotency of either parent within the union. 

8 See * Phil. Trans.,' A, vol, 187, p. 270. 

^ See 'Roy. Soc. Proc.,' vol. 62, p. 413. The paternal correlations, for reasons stated in the paper, are 
excluded from the result. 

^ See p. lis et seq. of the present memoir. 

^ See p. 98 et seq. of the present memoir. The grandparental correlation is based on two cases only. 

7 See *Roy. Soc. Proc.,' vol. 66, p. 140 et seq. 

^ Sec * Roy. Soc. Proc.,* vol. 65, p. 154. I have deduced the value for parents and grandparents from 
Dr. Warren's results for midparent and midgrandparent. The value for whole siblings I obtained from 
Dr. Warren's measurements, which he with great kindness placed at my disposal. 

(14.) Conclusions. — The course of this investigation has not been without diffi- 
culties, and I am fully prepared to admit that more obscurity and greater probable 
errors are likely to arise when we deal with the inheritance of a character not directly 
measurable, than when we take that of a character to which we can at once apply a 
quantitative scale. But I contend that many of the characters, the inheritance of 
which it is most important to investigate, do not at present, and perhaps never will, 
admit of a quantitative measurement. We can arrange in order, we can classify, we 
can say more or less intense, but we cannot read off value on a scale. It is just such 
characters also, which the not highly trained observer can most easily appreciate and 
record. Hence we have been compelled to devise some method of dealing with them, 
and the present paper illustrates how the method invented can be applied to reach 
results of considerable interest and of substantial validity. 



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120 PROFESSOR K. PEARSON AND DR. A. LEE ON 

In order to illustrate the method, I chose two characters, coat-colour in horses and 
eye-colour in man, which seemed sufficiently diverse both as to origin and species.* 
The new method enabled me to reach results for half-brethren, grandparents and 
uncles and aunts, which had not yet been independently considered. The conclu- 
sions arrived at for eye-colour in man at no point conflict with those for coat-colour 
in horses, and both in the main accord with the theory of exclusive inheritance with- 
out reversion herein developed. We find — 

(i.) No approach to a single value for the coefficient of inheritance for each grade 
of relationship; it varies widely with the sex, and the line through which the 
relationship is traced. 

(ii.) No approach in average values to those which would be indicated by Mr. 
Galton's Law. 

Nor does the modification of Mr. Galton's Law, which I have termed the Law of 
Ancestral Heredity, give better results. For, if we cause it to give the parental 
values, it then renders results inconsistent with the fraternal values. 

(iii.) There is agreement with the theory of exclusive inheritance without reversion 
for the parental, avuncular and fraternal series ; but there is some anomaly in the 
case of grandparental inheritance. This requires further investigation, and possibly 
a modification of our views on the nature of reversion. 

We want a list formed of characters in various types of life, which are supposed to 
be exclusively inherited, and then experiments ought to be made and statistics col- 
lected with regard to these characters. It is in this field of exclusive inheritance 
that we must look for real light on the problem of reversion. 

If we consider the three known forms of inheritance, the blended, the exclusive, 
and the particulate (which may possibly be combined in one individual, if we deal 
with different organs) ; if we consider further that these forms may possibly have to 
be supplemented by others not yet recognised {e.g.y reversional theories depending, 
say, on heterogamous unions), then it would appear that the time is hardly ripe even 
for provisional mechanical theories of heredity. What we require to know first is, 
the class of organs and the types of life for which one or other form of inheritance 
predominates. As variation in no wise depends on the existence of two germ-plasms, 
so biparental heredity can by no means be treated as the result of their simple quanti- 
tative mixture ; the component parts of these germ-plasms corresponding to special 
characters and organs, must be able to act upon each other in a variety of qualita- 
tively different ways. To adopt for a moment the language of Darwin's theory of 
pangenesis, the multiplying gemmules from an organ in the father must (i.) cross with 
gemmules from that organ in the mother, and the hybrid gemmules give rise to 
blended inheritance, (ii.) must without crossing multiply alongside the gemmules of 
the mother, and give rise to particulate inheritance, (iii.) must alone survive, or alone 

* Since supplemented by my investigations, based on Mr. Galton's data, for coat-colour in hpunds, 
* Roy. Soc. Proc.,' vol. 66, p. 140 et seq. 



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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 121 

be destroyed in a struggle for existence with those of the mother, and give rise to 
exclusive inheritance. And all these three processes may be going on within the 
same germ-plasm mixture at the same time ! Even without using the language of 
gemmules, processes analogous to the above must be supposed to take place. Thus a 
quantitative " mixture of germ-plasms " becomes a mere name, screening a whole 
range of mechanical processes ; and very possibly a new one could be found for each 
new form of heredity as it occurs. Such processes like the old ones would still 
remain without demonstrable reality under the veil of " mixture of germ-plasms." 

What I venture to think we require at present is not a hypothetical plasmic 
mechanics, but careful classifications of inheritance for the several grades of rela- 
tionship, for a great variety of characters, and for many types of life. This will 
require not only the formation of records and extensive breeding experiments, but 
ultimately statistics and most laborious arithmetic. Till we know what class o^ 
characters blend, and what class of characters is mutually exclusive, we have not 
within our cognizance the veriest outlines of the phenomena which the inventors of 
plasmic mechanisms are in such haste to account for. Such inventors are like planetary 
theorists rushing to prescribe a law of attraction for planets, the very orbital forms of 
which they have not first ascertained and described. Without the observations of 
Tycho Bbahe, followed by the arithmetic of Kepler, no Newton had been possible. 
The numerical laws for the intensity of inheritance must first be discovered from wide 
observation before plasmic mechanics can be anything but the purest hypothetical 
speculation. 



Appendix I. 

Tables of Colour Inheritance in Thoroughbred Racehorses, extracted by Mr. Leslie 
Bramley-Moore jfrom Weatherby's Studbooks. 



Table of Colours. 

1 = black (bl.) 9 = chestnut or bay (cli./b.). 

2 = black or brown (bl./br.). 10 = chestnut (ch.). 

3 = brown or black (br./bL). 11 = chestnut or roan (ch./ro.). 

4 = brown (br.). 12 = roan or chestnut (ro./ch.). 

5 = brown or bay (br./b.). 13 = roan (ro.). 

6 = bay or brown (b./br.). 14 = roan or grey (ro./gr.). 

7 = bay (b.). 15 = grey or roan (gr./ro.) 

8 = bay or chestnut (b./ch.). 16 = grey (gr.). 



VOL. CXOV. — A. R 



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f-H 


CO 
rH 


1 






































































Eh 



B 2 



Digitized by VjOOQ IC 



124 



PEOFESSOE K. PEARSON AND DB. A. LEE ON 

CoUb. 



i 
1 
J 



1^ 

•rj tJ 



I 





J 


CO 

rH 


*- 


I-H 


5 

r-H 


lO 


CO 


s 


o 


o 


i 


o 


o 


o 


o 


t—i 


01 


r^ 




CO 


& 


o 


o 


o 


o 


o 


o 


^ 


o 


o 


CO 


o 


o 


o 


o 


o 


o 


t^ 




r-H 




o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




rH 




o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




I-H 


s 


o 


o 


o 


o 


o 


o 


I-H 


o 


o 


<N 


o 


o 


o 


o 


^H 


o 


-* 




r-H 




o 


o 


o 


o 


o 


o 


r^ 


o 


o 


o 


o 


o 


o 


o 


o 


o 


rH 




r-l 




o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


1^ 


r-i 




o 

I-H 


4 


'^ 


o 


o 


00 

f-H 


I-H 


CO 


CO 

00 


o 


o 


eo 

CO 

I-H 


o 


o 


o 


o 


o 


r-i 


5 




o> 




o 


o 


o 


o 


o 


o 


t—i 


o 


o 


o 


o 


o 


o 


o 


o 


o 


I-H 


1 


00 


t 
^ 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




^ 


^ 


eo 


^ 


o 


o 

CO 


I-H 


I-H 




o 


o 


r-H 


o 


o 


o 


o 


o 


o 






<o 


1. 


o 


o 


o 


lO 


o 


CO 


CO 


o 


o 


cq 


o 


o 


o 


o 


o 


o 


CO 

r-H 




»o 


e 
^ 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




'^ 


i 


eo 


cq 


o 




eo 


I-H 


CO 


o 


o 


t- 

C9 


o 
o 


o 
o 


o 


o 


o 


o 






CO 


^ 
i 


o 


o 


I-H 


o 


o 


o 


o 


o 


o 


o ^ 


o 


o 


o 


o 


t-* 




C<I 


4 

3 


I-H 


o 


o 


r^ 


o 


o 


I-H 


o 


o 


o 


o 


o 


o 


o 


o 


o 


eo 




l-l 


3 


lO 


rH 


o 


CO 


o 


r^ 


o> 


o 


o 


CO 

4 


o 
2 


o 


o 

g 


o 


o 


o 


00 






:i 


1- 

3 




M 


e 
i 




^ 




e 
4 


& 
? 


4 

bo 


^ 


: 






r^ 


<N 


CO 


^ 


iO 


CO 


t* 


00 


a> 


o 

r-H 


r-H 
I-H 




eo 


I-H 


I-H 


CO 

I-H 


i 



Digitized by VjOOQ IC 



MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 125 

FiUies. 



J 
I 
i 



8 p 
'g 3 



S 



a 





o 




o 


lO 


I-H 


CO 


o 


«o 


^ 


^-1 


o 


a> 


o 


o 


rH 


rH 


Ci| 


t* 








i-« 


^-c 




''I* 




lO 


<o 






a> 














o 












r^ 






-**< 






cs 














o 




H 








































• 
r-l 


& 


o 


o 


o 


o 


o 


o 


rH 


o 


o 


CO 


o 


o 


o 


o 


o 


CO 


t* 




to 


2 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




I-H 


& 






































^ 


i> 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




r^ 


S 






































CO 

I-H 


2 


o 


o 


o 


rH 


o 


o 


rH 


o 


o 


o 


o 


o 


r^ 


I-H 


o 


o 


-* 






M 






































<M 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


^-1 


o 


o 


o 


o 


o 


o 


I-H 




I-H 


t 






































rH 


2 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




r«H 


-g 






































d 


.d 


c^ 


O^l 


o 


00 


o 


<o 


a> 


r^ 


o 


55 


o 


o 


o 


o 


o 


cq 


CO 




i-M 


t) 








I-H 






l>- 






^-1 














§ 




o> 


,0 


o 


o 


o 


o 


o 


o 


o 


o 


o 


f-l 


o 


o 


o 


o 


o 


o 


rH 






" 




































^ 


00 


:« 
^ 


o 


o 


o 


o 


o 


o 


r^ 


o 


o 


o 


o 


o 


o 


o 


o 


o 


r^ 




. 


^ 


^ 


00 


o 


t- 


o 


r-H 


00 


o 


o 


pi 


o 


o 


o 


o 


r^ 


r^ 


o. 






fjQ 








CD 




CO 


00 
C9 






o 

I-H 














§ 




<d 




i-H 


o 


r-l 


e<9 


o 


I-M 


CO 


o 


o 


l>- 


o 


o 


o 


o 


o 


o 


00 

I-H 




id 




o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




'^ 


»: 


I-H 


'^ 


o 


c^ 


o 


lO 


^ 


o 


o 


t* 


o 


o 


o 


o 


r^ 


I-H 


lO 




^ 








»o 




rH 


l>- 






<N 




















F^ 






































CO 


i 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




c<i 


<! 
3 


I-H 


o 


o 


o 


o 


o 


I-H 


o 


o 


f-H 


o 


o 


o 


o 


o 


o 


CO 






,^ 


I-H 


l-l 


o 


CO 


o 


CO 


CO 


o 


o 


-**< 


o 


o 


o 


o 


o 


o 


o 






3 














^-c 




















(M 






3 


3 


^ 

i 


i 






rd 


1 

2" 




■i 


2 


s 


2 


-1 

2 


2^ 


& 


: 








I-H 


(M 


CO 


-**< 


lO 


o 


t^ 


00 


o> 


o 


I-H 
I-H 


I-H 


CO 

r-H 


'* 


^-1 


CO 


H 



Digitized by VjOOQ IC 



126 



PROFESSOR K. PEARSON AND DR. A. LEE ON 

CoUs. 



i 
I 

1 3 

^ i 

i I 

•I I 

-« i 



n 

e3 





1 


00 


CO 


'^i^ 


CD 


CO 


2g 


tr 


o 


o 


•^ 


o 


o 


f-< 


I-H 


o 


lO 






i-i 


r^ 




I-H 




^ 






s 














§ 




































f-H 




1-4 


& 


o 


o 


o 


I-H 


o 


o 


pH 


o 


o 


I-H 


o 


o 


o 


o 


o- 


r-H 


-**< 






d 






































o 


u 


o 


o 


o 


o 


o 


o 


O 


o 


o 


o 


o 


o 


o 


o • 


o 


o 


o 




r-H 


fe 


























o 












'^ 


.& 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




rH 


2 






































CO 


2 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 






,4 






































(M 


4j^ 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




rH 


2 






































I-H 


2 


o 


o 


o 


o 


o 


o 


o 


o 


o 


(M 


o 


o 


o 


o 


o 


o 


<M 




r-H 


4 






































o 


4 


-«»« 


o 


r-H 


a> 


o 


C<l 


o 


o 


o 


00 


o 


o 


o 


o 


o 


I-H 


to 




r^ 








G^ 




I-H 


o 






(M 














t* 




















^H 






r-H 








o 






<N 




O) 


4 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


f 




t) 




































] 


00 




o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 






^ 


(M 


I-H 


C<l 


-* 


f-H 


'^i^ 


«o 


o 


o 


CO 


o 


o 


1-H 


r^ 


o 


<N 


^ 




t^ 


rfi 


r-H 


I-i 




Oi 




(M 


00 
(M 






I-H 














2 




<o 


4 
^ 


I-H 


o 


o 


C<l 


o 


CO 


00 


o 


o 


I-H 


o 


o 


o 


o 


o 


I-H 


to 

I-H 




to 




o 

r-H 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 






gl 


C<1 


I-H 


a> 


(M 


t^ 


r-H 


o 


o 


t* 


o 


o 


o 


o 


o 


o 


o 






il 








(M 






kO 






I-H 














I-H 

I-H 






^ 






































CO 




o 


o 


o 


O 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




(M 


i 

3 


o 


o 


o 


o 


o 


o 


I-H 


o 


o 


C<I 


o 


o 


o 


o 


o 


o 


CO 




I-H 


3 


o 


o 


o 


I-H 


o 


(M 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


CO 






3 


4 

3 




i 


4 


^ 

e 
^ 


^ 




4 


o 


g^ 

^ 




2 




1^ 


^ 


: 








i-H 


« 


CO 


'^ 


\Q 


CD 


t* 


00 


a 


r-H 


(M 


CO 


-^ 


»o 


(D 


i 


























""^ 


*"* 












o 








































H 



Digitized by VjOOQ IC 



MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 127 

Fillies. 



•I 

i 

1 I 



4 



o 



O 

I 

I 

1 1 



3 













































■i! 




o 


Oi 


CO 


r-H 


r^ 


00 


o 


l-H 


o 


t^ 


r^ 


o 


C<l 


o 


r-H 


CO 


^ 




S 




CO 






CO 




-* 


t^ 






O) 


















o 










I-H 






^ 






<N 














^ 




H 




































r-i 




CO 

rH 


& 


r-H 


o 


o 


o 


o 


o 


o 


o 


o 


r-H 


o 


o 


o 


o 


o 


(N 


^ 






d 




































to 




o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 






^ 






































^ 


^ 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o ' 


o 


o 






g 






































CO 


2 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 






^ 


































^-1 




o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




I-H 


-2 


o 


o 


o 


o 


o 


o 


o 


o 


o 


I-H 


o 


o 


o 


o 


o 


o 


r-H 






-g 






































d 


,d 


00 


T-t 


(M 


I-H 


o 


t* 


I-H 


o 


o 


CO 


o 


o 


r^ 


o 


o 


C<l 


o> 




f-H 


o 








CO 






o 

I-H 






•-H 














CO 




O^ 


e 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


t? 




■€ 












































































^ 




^' 




































g 


00 


;«_ 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


cs 




^ 






































^<« 


^ 


CO 


CO 


r^ 


(M 


r^ 


r-^ 


lO 


r-l 


o 


r^ 


r-H 


o 


rH 


o 


r^ 


cq 


Oi 






r-H 






l>- 




CO 


s 






r-H 














g 




<p 


4 

^ 


o 


o 


o 


'^ 


o 


CO 


kO 


o 


o 


CO 


o 


o 


o 


o 


o 


o 






ko 


4 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




^ 


i 


lO 


01 


o 


C<4 


o 


CO 


s 


o 


o 


s 


o 


o 


o 


o 


o 


o 


Ol 






• 






































CO 


5. 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 






fj 






































« 


3 


o 


o 


o 


C9 


o 


r-H 


CO 


o 


o 


I-i 


o 


o 


o 


o 


o 


o 


t* 




f-H 


3 


o 


o 


o 


o 


o 


o 


l-l 


o 


o 


o 


o 


o 


o 


o 


o 


o 


r^ 






3 


u 

e 
s 


^ 
i 


rS 




^ 

^ 


^ 






4 


g 
3" 


4 


s 




i. 
^ 


& 


: 








rH 


c^ 


CO 


-**< 


lO 


CO 


t* 


00 


pi 


o 


I-H 


C4 


CO 


^ 


lO 


CO 


• 


























""* 


"^ 


"^ 


'^ 


*"* 


"^ 











































H 



Digitized by VjOOQ IC 



128 



PROFESSOR K. PEARSON AND DR. A. LEE ON 

Second CoU. 



s 




(D 




M 




■^ 




o 




u 




W 




^ 


CO 


3 


£ 


w 


CO 








^ 




a 


d 


2i 


So 


sS 


^ 


T 


Q 


■+2 


.^ 


3 


s 


rO 


pC4 


S 



I 



(D 



•s 


OQ 


1 


.^ 


'1 


a 


© 




•4^ 


Ti 


^ 


§ 


<? 


,^ 


1, 


a 


g 




H 







a 
1 




CO 


eo 


c^ 


00 


r-^ 


CI 


-^ 


o 


I-H 


CD 


o 


o 


C9 


o 


o 


00 


^ 






(M 


r-H 




CD 

I-H 




^ 


!§ 






00 














§ 




H 




































r^ 




r-H 


^ 


o 


o 


o 


o 


o 


o 


I-H 


o 


o 


o 


o 


o 


o 


o 


o 


C9 


CO 




id 


2 


o 


o 


o 


I-H 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


I-H 




1— 1 


& 






































I-H 


2 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




eo 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




T-H 


h 


— 







































M 


































I-H 




o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 






d 




































^-C 


(1 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




l-l 


4 






































d 


rd 


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r*i 


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eo 


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rH 


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r-H 
I-H 






oo 














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CO 




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J 


































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^ 


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^ 






































t^ 


^ 


00 


>o 


r-H 


(N 


o 


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GS 


o 


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o 


o 


r-i 


o 


o 


<M 


o 










t^ 




V— 4 


'^ 






I-H 














t^ 




















(M 






r-H 














^ 




CO 


.1. 
rf5 


1— 1 


o 


I-H 


o 


o 


o 

I-H 


pi 

r-H 


o 


o 


to 


o 


o 


o 


o 


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^ 




kd 




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o 


o 


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o 


I-H 


CO 




x> 








"^ 






CD 






ei 














^ 








































f-H 






.-i 






































eo 




o 


o 


o 


o 


o 


o 


cq 


o 


o 


o 


o 


o 


o 


o 


o 


o 


cq 




c<i 


4 

3 


I-H 


o 


o 


CO 


o 


o 


CO 


o 


o 


<N 


o 


o 


o 


o 


o 


o 


04 




1-3 


IB 


C<l 


o 


o 


'^ 


o 


r-H 


id 


o 


o 


C<l 


o 


o 


o 


o 


o 


o 


1^ 








3 


e 


s. 


M 


^ 

ij 


^ 


rd 


:S 


4 


4 


2^ 


i 


g 




^ 


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i 




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to 


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o 

I-H 


I-H 


C4 

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1-H 


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CD 

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i 



Digitized by VjOOQ IC 



MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 129 



I 

S 2 

1 i 

W 

6 i 



- 


1 


co 




-«*< 


CO 


c^ 


s 




o 


I-I 




o 


o 


cq 


o 


iH 


rH 

I-I 


i 




rH 


& 


•o 


o 


o 


^-c 


o 


o 


CO 


o 


o 


CO 


o 


o 


o 


o 


o 


^ 


r^ 




to 

1— 1 


4. 


o 


o 


o 


r^ 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


IH 




o 


o 


o 


o 


o 


o 


o - 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




CO 

I-H 


g 


o- 


o 


o 


o 


o 


o 


f-l 


o 


o 


I-I 


o 


o 


■ o 


o 


o 


o 


« 




c4 


1 


o 


o 


o 


o 


o 


o 


o 


o 


o 


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o 


o 


o 


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o 


o 




l-H 

rH 


4 


o 


o 


o 


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o 


o 


o 


o 


o 


o 


o 


o 


o- 


cr- 


"O 


o 


o 






-i 


t^ 


00 


o 


2 


1-1 


CO 

T-t 


cq 


o 


o 


oq 


o 


o 


iH 


o 


o 


CO 


s 

kO 


1 


o> 




o 


o 


o 


o 


o 


o 


I-H 


o 


o 


o 


o 


o 


o 


o 


o 


o 


iH 


! 




1 


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o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


i 


CO 


00 


CO 


1-^ 


r-i 


s§ 


1 


o 


1-^ 




o 


o 


I-H 


o 


o 


CO 




CD 




C<l 


o 


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o 


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s 


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o 


CO 


o 


o 


o 


o 


o 


o 


00 
00 




id 


■^ 

i 


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1— 1 


o 


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rH 


o 


o 


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o 


o 


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cq 




^ 


^ 


o 

I-i 


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■ o 


g 


o 


I-H 




o 


o 


00 


o 


o 


o 


o 


iH 


fH 


IH 

CO 




CO 


A. 

3 


o 


o 
o 


o 
o 


o 


o 
o 


I-I 


CO 


o 


o 


o 


o 


o 


o 


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o 


o 


-* 




kO 


o 


00 


o 


o 


00 


o 


o 


o 


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o 






• 


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^ 


■l-l 


o 


o 


o 


oq 


eo 
1— 1 


o 


o 


t* 


o 


o 


o 


o 


o 


o 


s 






3 


4 

3 




M 


^ 
i 


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^ 


^ 




4 


4 


-8 




g 




1^ 


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f-H 


eq 


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^ 


la 


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t* 


00 


Oi 


o 

fH 


' »H 

IH 


iH 


eo 

FH 


-«*< 

r--* 


IH 


IH 


i 



VOL. CXOV. — A. 



Digitized by VjOOQ IC 



130 



PROFESSOR K. PEARSON AND DR. A. LEE ON 

Second Colt. 







1 




00 


-* 


(M 


00 


1-f 


CO 


00 


o 


o 


Oi 


o 


o 


r-l 


r^ 


o 


oq 


^ 








r-H 






CO 




CO 








00 














o 






O 




































^H 






H 




























o 












CO 


c> 


o 


o 


o 


o 


o 


o 


r^ 


o 


o 


o 


o 


o 


o 


o 


cq 


CO 1 






r-< 


to 


































1 




d 


l 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


1 








^ 


































1 


,,-^ 


1— < 


^ 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


00 






o 




































frt 






t-, 




































4) 








































M 










































1 




CO 


2 


o 


o 


o 


1-f 


o 


o 


o 


o 


o 


f^ 


o 


o 


o 


o 


o 


o 


<N 


u 










































pq 












































I-H 


t 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


i 
o 1 








2 




































§ 


T-t 


2 


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o 


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I-H 






































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02 




-s 




































© 








































So 


^ 


d 

I-H 


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o 


o 


o 


g 


o 


CO 


I-H 

00 


o 


o 


CO 


o 


o 


o 


o 


o 


o 


00 




























1— 1 














oq 




© 22 

^ i 








































ci 




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o 


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o 


o 


o 


o 


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o 


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►> 


00 




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o 


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o 


o 


."tj 






^ 




























o 


o 


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CO 


, 


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00 


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f-H 


It- 


o 


t^ 


o 


o 


o 


Oi 


o 


o 


r-l 


w 


6 


t^ 








CO 




l-l 








00 














ij 






. 




































2 


g 


<d 


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o 


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o 


oq 


00 


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o 


CO 


i 

1 


-2 

3 




rJCJ 




































id 


e 
i 


o 


o 


o 


o 


rH 


o 


o 


o 


o 


o 


o 


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o 


o 


o 


o 


r^ 


1 

H 

•— t 




^* 


i 


t* 


CO 


o 


CO 


o 






o 


o 




o 


o 


o 


I-H 


o 


o 


§8 


P> 










































1 


CO 




o 


f-H 


o 


o 


o 


o 


1— 1 


o 


o 


o 


o 


o 


o 


o 


o 


o 


c\ 


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pO 




































EH 




fi^ 








































d 


3 


o 


o 


o 


T-t 


o 


o 


-**< 


o 


o 


c^ 


o 


o 


o 


o 


o 


o 


t* 




PH 


3 


CO 


o 


I-H 


kO 


o 


o 


o> 


o 


o 


ei 


o 


o 


o 


o 


o 


o 


o 
cq 






3 


t 


i 


i 


e 

i 


^ 

^ 


^ 




^ 
-§ 


4 


4_ 


t 
1 


i 




2 


^ 


: 








T-l 


C<1 


CO 


-* 


lO 


<o 


t* 


00 


o 


o 


rH 




CO 




r-l 


CO 


1 











































Digitized by VjOOQ IC 



MATHEMATICAL CONTMBUTIONS TO THE THEORY OF EVOLUTION. 131 



o 
Q P 



1 ^ 

■f3 06 

^3 



k 

^ 





1 


00 

eo 


1-4 


-* 


to 


C4 


C9 


I—* 


o 


o 


1 


o 


o 


CO 


T-t 


o 


lo 


(M 




to 

1-4 


& 


O 


o 


o 


o 


o 


o 


f-H 


o 


o 


o 


o 


o 


o 


o 


o 


-^ 


to 




to 

^-1 


-1 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




r-l 


2 


o 


o 


o 


^-c 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


1-4 






2 


o 


o 


o 


1-4 


o 


o 


1-4 


o 


o 


1-4 


o 


o 


o 


o 


o 


o 


CO 




c4 

1— 1 


1 
2 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




^-1 

I-H 


^ 
-i 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




d 

I-H 




©^ 


e«i 


o 




o 


t^ 




o 


o 


00 

eo 


o 


o 


l-l 


o 


o 


o 


to 


j 


oS 




o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


! 


00 


^2" 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


1 


t-* 


^' 


I—I 


-**< 


C<l 


5 


o 


s 




o 


o 




o 


o 


fH 


o 


o 


IH 




<d 


1. 


o 


o 


o 


1-i 


o 


-**< 


s 


o 


o 


t^ 


o 


o 


o 


o 


o 


o 


s 




to 


^ 
i 


o 


o 


o 


o 


« 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


(M 




^ 


i 


©^ 


'^i^ 


o 


I-H 


o 


T-t 


1—t 
1— 1 


o 


o 


iO 


o 


o 


f-H 


iH 


o 


o 


^ 

^ 




CO 


^ 
i 


rH 


1-4 


o 


o 


o 


o 


cq 


o 


o 


o 


o 


o 


o 


o 


o 


o 


•«* 




(M* 




o 


o 


rH 


^ 


o 


o 


-**< 


o 


o 


cq 


o 


o 


o 


o 


o 


o 


1—t 




rH 


;^ 


<o 


o 


f-l 




o 


o 


I-H 


o 


o 


C<l 


o 


o 


o 


o 


o 


o 


00 

CO 






3 


^ 

3 




i 


e 
i 


^ 

^ 


^ 


4 


^ 
4 


4 


4 


4 

2 


2 


2 


1 

& 


^ 


• 






I-H 


<m' 


PS 


'^i^ 


o 


<o 


t^ 


00 


o> 


o 

I-H 


I-H 




eo 

rH 




IH 


to 


1 



s 2 



Digitized by VjOOQ IC 



.132 



PROFESSOR K. PEARSON AND DR. A- LEE ON 

Second FiUy. 



s .§ 



^ 


QQ 


•S 


■y 


QQ 


§ 


% 


jS 


w 


''S 




-4^ 




P5 


t^ 


rO 


■*s 


© 


•"O 


^ 


g 


1^ 


W 


^ 


'3 


y 


tri 


r/) 


-^ 






i 


9 


?3 


1 


PR 


M 


13 


M 


g 




Gi 


t3 


S 


PC 




H 


P^ 





4 




^ 


00 


rH 


T-^ 


o 


o 


CO 


rH 


o 


00 


r^ 


o 


^ 


o 


cq 


^« 










oq 






^ 




^ 


r^ 






lO 


















o 










rH 






lO 






C<l 














o 




EH 




































r-i 






& 


O 


rH 


o 


o 


o 


o 


(M 


o 


o 


1^ 


o 


o 


o 


o 


o 


r^ 


to 




lO 


2 


O 


O 


o 


o 


o 


o 


o 


o 


o 


rH 


o 


o 


o 


o 


o 


o 


r^ 






^ 










































o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




CO 


g 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 






^ 










































o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




• 
r-l 


g 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




r-l 


-^ 






































d 


rd 


-* 


CO 


1^ 


lO 


o 


*- 


CO 


r'H 


o 


a> 


o 


o 


<M 


o 


o 


f^ 


CD 
CO 




rH 


«? 








c^ 






o 




















(M 




0!» 


^ 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


-^ 








































•g 








































00 


-§ 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


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o 


o 


o 


o 


o 




iS" 






































t^ 


^ 


CO 

r-l 


CO 


o 


CD 


o 


CO 
01 


CO 

^^ 
CO 


o 


o 




rH 


o 


C9 


o 


r-^ 


cq 


CO 




«> 




rH 


o 


o 


lo 


o 


to 




o 


o 


o 


o 


o 


o 


o 


r^ 


o 


5 




id 




o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 




-* 


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'^ 


rH 


o 


SI 


o 


CO 


CO 


o 


o 


CO 


o 


o 


o 


o 


o 


CO 


1-^ 














































i-i 






































CO 




o 


o 


o 


o 


o 


o 


r^ 


o 


o 


o 


o 


o 


o 


o 


o 


o 


r^ 






M 






































c^ 


3 


o 


o 


o 


1^ 


o 


1-^ 


CO 


o 


o 


lO 


o 


o 


o 


o 


o 


o 


o 




I-H 


3 


(M 


o 


o 


^ 


o 


o 


o> 


o 


o 


o 


o 


o 


o 


o 


o 


o 


g 






3 


3 


§. 

u 

Xi 


i 




1. 


^ 


-i 
^ 


4 






g 


g 


g 


g^ 


^ 


: 








r-H 


<N 


CO 


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lO 


CO 


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Digitized by VjOOQ IC 



MATHEMATICAL C50NTBIBUTION8 TO THE THEORY OF EVOLUTION. 133 



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Digitized by VjOOQ IC 



134 



PEOFESSOE K. PEAESON AND DE. A. LEE ON 

Second Filly. 



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Digitized by VjOOQ IC 



MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 135 



e 

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2 
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Digitized by VjOQQ IC 



136 



PROFESSOR K. PEARSON AND DR. A. LEE ON 

Colts. 



u 

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.a -g 

^ I 

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v- CO 



>., OB 

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gr./ro. 


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ro./ch. 


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ch./ro. 


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ch./b. 


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b./ch. 


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br. br./b. 


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br./bl. 


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1 



Digitized by VjOOQ IC 



MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 137 

Cdis. 



GQ 



^1 



1^ 
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16. 
gr./ro. 


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14. 

i-o./gr. 


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10. 11. 

ch. ch./ra 


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br./bl. 


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3 


1, 

3 


} 


i 




4 


^ 


t 

^ 


4 




1 

■i 


4 

1 


g 


2 


^ 
^ 


& 


: 




i-H 


eq 


eo 


^ 


lO 


CO 


t^ 


00 


Oi 


o 

I-H 


I-H 
I-H 


C9 

I-H 


CO 

I-H 


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I-H 


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I-H 


i 



VOL CXCV. — A. 



Digitized by VjOOQ IC 



138 



PROFESSOR K. PEARSON AND DR. A. LEE ON 



Appendix II. 

Tables of Eye-colour Inheritance in Man, extracted by Kabl Pearson from 
Mr. Francis Galton's Family Records. 



1 = light blue. 

2 = blue, dark blue. 

3 = blue-green, grey. 

4 = dark grey, hazel. 



Table op Tints. 

5 = light brown. 

6 = brown. 

7 = dark brown. 

8 = very dark brown, black. 



This grouping is not quite in keeping with more recent divisions of eye-colour, but 
being that adopted by Mr, Galton in his original collection of data, it could not be 
modified in accordance with present practice. 

Tables for the Direct Inheritance of Eye-colour. First Generation. 

I. — Fathers and Sons. 1000 Cases. 

Failiers. 



GQ 



Tint. 


1. 


2. 


3, 


4. i 


5. 


6. 


7. 


8. 


Totals. 


1 


9 


12 


5 


5 , 





1 


2 





34 


2 


10 


163 


65 


36 ' 


1 


7 


15 


4 


301 


3 


10 


73 


124 


41 


1 


12 


18 


6 


284 


4 


4 


21 


34 


55 





11 


11 


1 


137 


5 











1 


2 


2 


1 





5 


6 


1 


26 


12 


19 ' 


1 


19 


16 


6 


100 


7 


1 


23 


16 


14 , 





11 


31 


2 


98 


8 


1 


4 


8 


10 





1 


7 


10 


41 


Totals 


36 


322 


264 


180 ; 

i 


5 


64 


101 


28 


1000 



II. — Fathers and Daughters. 1000 Cases. 
Faihers, 





Tint 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


Totals. 




1 


4 


9 


11 


4 





1 


2 


1 


32 


'n 


2 


11 


139 


57 


31 





6 


24 


6 


273 


^ 


3 


9 


73 


111 


38 


1 


15 


19 


3 


269 


1 


4 


5 


43 


34 


54 


2 


10 


14 


3 


165 


5 





1 


3 








3 


1 





8 


M 


6 


1 


45 


13 


19 





23 


15 


3 


11? 




7 


2 


27 


10 


12 





7 


41 


6 


105 




8 





8 


4 








2 


11 


4 


29 




Totals 


32 


345 


243 


158 


3 


67 


127 


25 


1000 



Digitized by 



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MATHEMATICAL CONTRIBUTIONS TO THE THEOEY OP EVOLUTION. 



139 



in. — Mothers and Sona 1000 Casea 
Mothers. 



Tint. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


Totals. 


1 


5 


14 


6 


3 





1 





6 


35 


2 


12 


119 


83 


29 


8 


20 


21 


9 


301 


3 


13 


54 


113 


35 


4 


37 


14 


8 


278 


4 


3 


21 


26 


54 


1 


17 


6 


6 


134 


5 





1 


1 








3 








5 


6 


1 


9 


26 


10 


1 


30 


24 


3 


104. 


7 


1 


9 


19 


16 





18 


31 


7 


101 


8 





7 


15 


4 


1 


3 


5 


7 


42 


Totals 


35 


234 


289 


151 


15 


129 


101 


46 


1000 






IV. — Mothers and Daughters. 1000 Cases. 
Mothers, 



Tints. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


■ 8. 


Totals. 


1 


5 


15 


3 


2 


2 


2 


2 





31 


2 


7 


99 


67 


29 


2 


15 


23 


13 


255 


3 


7 


77 


111 


38 


1 


26 


14 


6 


280 


4 


5 


22 


34 


46 


2 


27 


21 


7 


164 


5 





2 


2 





3 


1 


2 


1 


11 


6 





13 


27 


20 


1 


35 


17 


7 


120 


7 





13 


21 


16 


1 


19 


26 


9 


105 


8 


1 


5 


7 


2 


1 


4 


12 


2 


34 


Totals. 


25 


246 


272 


153 


13 


129 


117 


45 


1000 



Tables for the Collateral Inheritance of Eye-colour. 
V*.— Brothers and Brothers. 1500 Cases. 
First Brother, 





1 

First. 

1 


1. 


2. 


3. 


4. 


5. 


6. 


7 


8. 


Totals. 

1 




8 


2 


3 


4 














19 


1 


2 


36 


202 


23 


17 





6 


4 


3 


291 


3 


16 


182 


209 


26 





4 


2 


2 


441 


1 


4 


6 


36 


71 


84 





7 


2 





206 


5 





3 


2 


1 





1 








7 


f 


6 


3 


56 


50 


39 





34 


5 


6 


193 


fc 


7 


6 


37 


76 


48 


1 


36 


36 


2 


242 




8 


4 


24 


26 


18 





8 


6 


15 


101 




Totals 


79 


542 


460 


237 


1 


96 


55 


30 


1500 



T 2 



Digitized by 



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140 



PROFESSOR K. PEARSON AND DR. A. LEE ON 



OQ 



'^ 



VI*. — Sisters and Sisters. 1500" Cases. 
First Sister. 



First. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


1 

8. 


Totals. 


1 


10 


2 


1 


1 














14 


2 


17 


147 


29 


6 





10 


6 


2 


217 


3 


10 


136 


186 


24 





9 


5 


3 


373 


i 


3 


75 


94 


66 





1 


10 





249 


5 





3 


5 


2 


2 





1 





13 


6 


2 


57 


69 


55 


5 


62 


9 


2 


251 


7 


4 


56 


61 


52 


10 


59 


49 





291 


8 


2 


20 


10 


7 


6 


13 


26 


8 


92 


Totals 


48 


496 


455 


213 


23 


144 


106 


15 


1500 



V^ — Brothers and Brothers. Symmetrical System. 



Tint. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


Totals. 


1 


16 


38 


19 


10 





3 


6 


6 


98 


2 


38 


404 


205 


53 


3 


62 


41 


27 


833 


3 


19 


205 


418 


97 


2 


64 


78 


28 


901 


4 


10. 


63 


97 


168 


1 


46 


50 


18 


443 


6 





3 


2 


1 





1 


1 





8 


6 


3 


62 


54 


46 


1 


68 


41 


14 


289 


7 


6 


41 


78 


50 


1 


41 


72 


8 


297 


8 


6 


27 


28 


18 





14 


8 


30 


131 


Totals 


98 


833 


901 


443 


8 


289 


297 


131 


3000 



VI^ — Sisters and Sisters. Symmetrical System. 



Tint. 


1 

1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


Totals. 


1 


20 


19 


11 


4 





2 


4 


2 


62 • 


2 


19 


294 


165 


81 


3 


67 


62 


22 


713 


3 


11 


165 


372 


118 


5 


78 


66 


13 


828 


4 


4 


81 


118 


132 


2 


56 


62 


7 


462 


5 ! 





3 


5 


2 


4 


6 


11 


6 


36 


6 


2 


67 


78 


56 


5 


104 


68 


15 


396 


7 


4 


62 


66 


62 


11 


68 


98 


26 


397 


8 


2 


22 


13 


7 


6 


15 


26 


16 


107 


Totals 

1 


62 


713 


828 


462 


36 


395 


397 


107 


3000 



Digitized by 



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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 



141 



<S 



VII, — Brothers and Sisters. 1500 Cases. 
Brcihers. 





Tint. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


Totals. 




1 


5 


9 


18 


4 








1 





37 




2 


20 


163 


101 


36 





28 


19 


13 


380 


*5 


3 


9 


98 


193 


50 





37 


17 


14 


418 


■S 


4 


5 


36 


49 


67 


3 


28 


13 


16 


217 


il ' 5 





2 


5 


1 


1 


2 


2 


3 


16 




6 


3 


47 


41 


27 


4 


42 


17 


14- 1 


195 




7 


4 


34 


49 


22 


3 


27 


30 


19 ! 


178 




8 


1 

1 


10 


7 


1 





10 


8 


22 ■ 


59 




; Totals 


47 


399 


463 


208 


11 


174 


107 


,1 1 

1 


1500 



Table for Assortative Mating in Eye-colour. 

Vni. — Husbands and Wives, 774 Cases. 

Husbands, 





Tint 


1. 


'■ 


3. 


4. 

1 


5. 


6. 


7. 


8. 


Totals. 




1 


2 


13 


4 


3 





1 


2 





25 




2 


6 


87 


42 


26 





16 


13 


6 


196 


1 


3 


6 


56 


93 


31 


1 


16 


11 


6 


220 


4 


4 


32 


35 


18 1 


1 


15 


6 


1 


1 112 


5 








6 


1 1 








1 





7 




6 


1 2 


38 


27 


10 ! 


1 


12 


10 


1 


101 




7 


1 5 


20 


28 


7 


1 


6 


12 


4 


83 




8 


! 2 


8 


8 


2 





2 


4 


4 


30 




Totals 


: 27 

1 


254 


242 


98 


4 


68 


59 


22 


774 



Tables for the Direct Inheritance of Eye-colour. Second Generation. 

IX. — Paternal Grandfather and Grandson. 765 Cases. 

Paternal Gramdfather, 



First. 


i 

! 1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


Totals. 


1 


4 


10 


3 








1 


3 





21 


2 


7 


115 


31 


20 


1 


6 


13 


3 


196 


3 


5 


64 


109 


21 





10 


22 


4 


235 


4 


2 


25 


40 


21 





9 


13 


6 


116 


6 











1 





1 








2 


6 





14 


32 


11 





15 


5 


2 


79 


7 


4 


16 


16 


9 


1 


11 


21 


2 


80 


8 





6 


5 








3 


16 


6 


36 


Totals 


i 22 


250 


236 


83 

• 


2 


66 


93 


23 


765 



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142 



PBOFBSSOE K. PEAESON AND DR. A. LEE ON 



X. — Paternal Grandfather and Granddaughter. 681 Cases. 

Paternal Grandfather, 





First. 


1. 


2. 


3. 


4. 


5. 


6. 


7. • 


8. 


Totals. 




1 


3 


6 


4 


5 





1 


1 ' 





20 




2 


2 


94 


32 


10 


2 


6 


16 


4 


166 


1 


3 


5 


67 


71 


17 





9 


20 


3 


192 


4 1 


4 


36 


33 


26 


1 


10 


9 


3 


121 


5 





3 


4 


1 





1 


2 


1 1 


12 


'^ 


6 


1 


16 


21 


11 





15 


6 


4 ' 


74 


1 


7 


3 


10 


20 


11 


1 


8 


15 


3 


71 


8 

i 





2 


5 


1 


1 


1 


10 


5 


25 




Totals 


18 


233 


190 


82 


5 


51 


79 


23 


681 



Table XI. — Maternal Grandfather and Grandson. 771 Cases. 

Maternal Grandfather. 



Tint. 1 

i 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


Totals. 


1 


3 


11 


3 


1 






1 




19 


2 


8 


113 


46 


22 


1 


13 


5 


3 


1 211 


3 


10 


87 


89 


12 





11 


8 


3 


220 


^ 1 
5 ! 

6 


4 


33 

1 
25 


35 

1 

25 


22 


— 


15 


6 


2 


117 

2 

84 


2 


7 





14 


7 


4 


7 


1 


22 


26 


6 


2 


9 


10 


4 


80 


8 


— 


12 


12 


6 


— 


4 


3 


1 


38 


Totals 

1 


28 


304 


237 


76 


3 


66 


40 


17 


771 



I' 



I 



Table XII, — Maternal Grandfather and Granddaughter. 687 Cases. 

Maiemal Grandfaiher, 



Tint. 


1. 

1 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


Totals. 

1 


1 


1 


3 


7 


2 






1 




14 


2 


8 


84 


35 


11 





13 


6 


2 


159 


3 


11 


67 


76 


18 





7 


15 


5 


199 


4 


7 


41 


40 


14 





15 


11 


5 


133 


6 


— 


5 


2 








2 


1 


— 


10 


6 


4 


21 


32 


1 





16 


5 


2 


81 


7 


4 


15 


14 


8 


1 


7 


19 


2 1 


70 


8 


1 


5 


5 


1 




1 


5 


4 1 


21 


Totals 


35 


241 


211 


55 


1 


61 


63 


20 ! 


687 



Digitized by 



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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 



143 



Table XIII. — Paternal Grandfather and Grandson. 741 Cases. 

Paternal Grandmother. 



1 

Tint. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


Totak. 


1 


1 


2 


7 


1 


1 


2 


3 




17 


2 


6 


62 


69 


22 


4 


15 


25 


4 


207 


3 


4 


31 


95 


22 


1 


25 


33 


9 


220 


4 


3 


18 


36 


20 


4 


15 


16 


4 


116 


5 


— 


— 





1 








1 


1 


3 


6 


1 


10 


23 


6 


2 


16 


10 


4 


72 


7 


3 


15 


10 


4 


1 


14 


13 


11 


71 


8 


1 


10 


3 


3 


— 


« 


5 


8 


36 


Totals 


19 


148 


243 


79 


13 


92 


106 


41 


741 



I" 



I 



Table XIV. — Paternal Grandmother and Granddaughter. 717 Cases. 

Paternal Grandmother. 



Tint. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


Totals. 


1 




3 


3 


2 






6 


2 


16 


2 


7 


53 


56 


14 


2 


13 


28 


8 


181 


3 


8 


35 


65 


22 


6 


29 


28 


7 


200 


4 


4 


29 


36 


20 


3 


16 


23 


8 


139 


5 


1 


— 


3 


1 





5 


1 


— 


11 


6 





9 


29 


2 


4 


27 


8 


3 


82 


7 


— 


10 


15 


12 


2 


10 


12 


7 


68 


8 


1 


2 


1 


2 


1 


- 4 


4 


5 


20 


Totals 


21 


141 


208 


75 


18 


104 


110 


40 


717 



I 



Table XV. — Maternal Grandmother and Grandson. 756 Cases. 

Maiemai OrcMdmotker. 



Tint. 


1. 


2. 


3. 


4. 


5. 


j 

6. 


7. 


8. 


Totals. 


1 


1 


10 


1 


3 




1 


3 


1 


19 


2 


10 


68 


53 


23 





24 


13 


13 


204 


3 


9 


39 


67 


38 





32 


23 


11 


219 


4 
5 
6 


3 


34 


19 

1 

11 


30 

1 
10 


— 


19 


8 


4 


117 
2 

84 


^ 


20 


z 


24 


18 


1 


7 


2 


9 


23 


11 


1 


17 


17 


— 


80 


8 


— 


4 


6 


5 


— 


'< 6 


' 


7 


31 


Totals 


1 25 

t 


184 


181 


121 


1 


122 


85 


37 


756 



Digitized by 



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144 



PROFESSOK K. PEARSON AND DR. A. LEE ON 



Table XVI. — Maternal Grandmother and Granddaughter. 739 Cases. 

Maternal Grandmother. 





Tint 


1 1. 


2. 


3. 


4. 


6. 


6. 


7. 


8. 


Totals. 


, 


1 


2 


16 














18 


;§ 


2 


7 


66 


34 


13 


— 


21 


15 


6 


162 


§* 


3 


12 


62 


55 


25 


1 


27 


23 


5 


210 


^ 


4 


6 


32 


36 


25 





23 


15 


7 


144 


i 


5 





1 


3 


2 





3 


3 




12 


1 


6 


1 


14 


21 


11 





27 


17 


2 


93 


7 


1 "■" 


19 


17 


7 


— 


16 


17 


3 


76 




8 


1 


5 


4 


-2 


— 


7 


3 


3 


24 




Totals 


28 


212 


170 


85 


1 


124 


93 


26 


739 



Tables for the Collateral Inheritance of Eye-colour. Second Degree. 

XVIL — Paternal Uncle and Nephew. 1290 Cases. 

Paternal Uncle, 





Tint. 


1 

1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


Total. 




1 


4 


10 


11 


6 


1 


4 


5 


2 


43 




2 


11 


136 


98 


40 





26 


48 


12 


371 


§ 


3 


8 


84 


157 


26 


1 


27 


54 


7 


364 


^ 


4 





29 


69 


36 


1 


19 


27 


12 


193 


5 





2 


1 








2 


1 





6 




6 


1 


31 


35 


7 


1 


30 


19 


3 


127 




7 


2 


21 


27 


24 


1 


13 


34 


11 


133 




8 





11 


7 


6 





10 


8 


11 


53 




Total 


26 


324 


405 


145 


5 


131 


196 


58 


1290 



XVIII. — Paternal Uncle and Niece. 1128 Cases. 
Paternal Uncle. 





Tint. 


1. 

1 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


Total. 




1 


2 


10 


6 


2 





1 


6 


2 


29 




2 


7 


85 


61 


27 





29 


26 


13 


248 


^^ 


3 


1 6 


82 


126 


29 


1 


26 


43 


7 


319 


•^ 


4 


2 


47 


73 


40 


1 


29 


40 


5 


237 


-e; 


5 


1 


8 


1 


4 





1 


5 


4 


24 




6 


1 


26 


35 


12 


1 


8 


42 


3 


128 




7 


: 1 


20 


26 


19 





22 


26 


7 


120 




8 


1 


4 


2 


3 





5 


3 


6 


23 




Total 


1 26 


282 


329 


136 


3 


121 


191 


47 


1128 



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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 



145 



Table XIX. — Maternal Uncle and Nephew. 1242 Cases. 

Maternal Unele. 





Tint. 


1. 


2. 


3. 


4. 


5. 


6. 


7 


8. 


Totals. 




1 


1 


8 


13 


3 





3 


4 


1 


33 




2 


17 


137 


71 


29 





19 


14 


9 


296 


g 


3 


10 


128 


153 


26 





29 


34 


3 


383 


1" 


4 


2 


50 


62 


28 





22 


14 


1 


179 I 


5 





1 














1 





2 




6 


4 


33 


29 


12 





35 


20 


3 


136 




7 


1 


33 


40 


11 





26 


27 


2 


140 




8 





9 


17 


23 





8 


3 


13 


73 




Totals 


35 


399 


385 


132 





142 


117 


32 


1242 i 



Table XX. — Maternal Uncle and Niece. 1434 Cases. 

Maternal Uncle. 



Tint. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


Totals. 


1 


2 


15 


9 


2 





2 


1 





31 


2 


6 


99 


76 


23 





18 


13 


11 


246 


3 


18 


100 


108 


23 





37 


36 


10 


332 


4 


4 


72 


64 


28 - 





16 


21 


9 


214 


5 





14 


2 


3 





8 


5 





32 


6 


5 


38 


41 


10 





23 


11 


4 


132 


7 


1 


27 


25 


7 





19 


14 


3 


96 


8 





15 


5 


6 





9 


11 


6 


51 


Totals 


36 


380 


330 


102 





132 


112 


42 


1134 



* 



Table XXI. — Paternal Aunt and Nephew. 1186 Cases. 

Paternal Awnt. 



Tints. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


Totals. 


1 


6 


13 


5 


3 


1 


4 


6 


4 


42 


2 


19 


113 


83 


45 





36 


29 


5 


330 


3 


10 


81 


147 


30 


— 


29 


35 


8 


340 


4 


8 


28 


66 


38 


— 


18 


22 


11 


191 


5 


.i__ 


__ 


._», 


_ 





— 


— 


^ 





6 


3 


23 


35 


12 


1 


35 


10 


5 


121 


7 


5 


22 


28 


19 





18 


16 


5 


112 


8 


1 


4 


9 


8 


— 


6 


4 


13 


47 


Totals 


52 


284 


373 


155 


2 


148 


121 


51 


1186 



VOL. CXCV, — A. 



V 



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146 



PBOFESSOR K. PEAESON AND DR A. LEE ON 



Table XXII. — Paternal Aunt and Niece. 1149 Cases. 

Paternal Aunt. 



Tints. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


Totals. 


1 


2 


11 


2 


3 


_ 


2 


11 


2 


33 


2 


16 


89 


62 


37 


2 


25 


40 


14 


284 


3 


12 


93 


119 


40 


3 


41 


26 


12 


346 


4 


10 


36 


62 


43 


5 


25 


21 


11 


213 


6 





5 


7 


— 


— 


1 


3 


— 


16 


6 


1 


24 


33 


16 


1 


29 


19 


5 


128 


7 


2 


20 


28 


12 


2 


10 


22 


4 


100 


8 


— 


7 


— 


4 


— 


5 


9 


4 


29 


Totals 


42 


28.5 


313 


155 


13 


138 


151 


62 


1149 



t 



Table XXIII. — Maternal Aunt and Nephew. 1145 Cases. 

Maternal Aunt. 



First. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


Totals. 


1 


4 


8 


7 


3 




2 


3 


1 


28 


2 


5 


117 


81 


29 





43 


29 


6 


310 


3 


1 


73 


132 


38 





57 


43 


3 


347 


4 


1 


20 


54 


27 


1 


21 


11 


— 


135 


5 








3 


2 


— 


1 


— 


— 


6 


6 


— 


24 


35 


22 


— 


30 


23 


3 


137 


7 


— 


26 


29 


20 


1 


26 


25 


8 


136 


8 


— 


14 


6 


10 


— 


12 


4 


2 


47 


Totals 


11 


282 


346 


151 


2 


192 


138 


23 


1145 



Table XXIV. — Maternal Aunt and Niece. 1058 Cases. 

Maternal Aunt, 



First. 


1. 


2. 


3. 


4. 


5. 


6. 


7. 


8. 


1 
Totals. 


1 


2 


3 


10 












15 


1 


6 


87 


86 


31 


— 


23 


14 


12 


; 258 


3 


3 


71 


125 


32 


1 


49 


41 


3 


1 326 


4 


— 


39 


51 


31 


1 


33 


19 


6 


, 180 


5 


— 


4 


6 


1 


— 


8 


5 


3 


! 27 


6 


1 


25 


47 


10 


— 


24 


9 


2 


' 118 


7 


— 


30 


29 


11 





14 


18 


10 


112 


8 


— 


5 


4 


5 


— 


2 


4 


3 


; 23 


Totals 


11 


264 


358 


121 


2 


163 


no 


39 


1058 



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MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 



147 



[Notes added July 3, 1900. 

Note I. Inheritance of Temper and Artistic Instinct. — In additional to the fraternal 
correlations given on p. 102, T have dealt with Mr. Galton's statistics for the inheri- 
tance of good and bad temper given in his 'Natural Inheritance' (p. 235). The 
following gives the distribution of good and bad temper among 1,294 brethren, as 
deduced by Mr. Yule. 

First Brother. 



I 
I 

Co 





Good Temper. 


Bad Temper. 


Totals. 


Good 
temper. 


330 


255 


585 


Bad 
temper. 


255 


454 


709 


Totals 


585 


709 


1294 



The correlation is '3167. 

A like table is that for artistic instinct in the direct line : — 

Parentage. 








Artistic. 


Non-artistic. 


Totals. 


1 


296 


173 


469 
1038 


"1 


372 


666 


Totals 


668 


839 


1507 



In this case the correlation is '4039. 

The fraternal correlation is somewhat low. The exact significance of the parental 
correlation is also somewhat vague, as the parentage is classified as artistic when 
one or both parents are artistic. But the two tables are very suggestive, they 
indicate how the new method will enable us to deal quantitatively even with 
characters like temper and artistic instinct to which it is impossible to apply directly 
a quantitative scale. With the introduction of a third or medium class, I believe it 
will be possible to obtain excellent results for heredity from very simple observations, 
and I have in hand at the present time a large series of observations on collateral 
heredity based upon such simple classifications. The reader should further consult 

u 2 



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148 



PROFESSOR K. PEARSON AND DR. A. LEE ON 



Mr. G. U. Yule's remarks on the association of temper and of artistic instinct in his 
memoir on '* Association/' ' Phil. Trans./ A, vol. 194, p. 290, 1900. 

Note IL On the Correlation of Fertility and Eye-Colour. — In the course of the 
present paper I have frequently referred to a probable influence of reproductive 
selection as the source of the progressive change in eye-colour, i.e., to a possibility that 
eye-colour is correlated with fertility. I saw from Mr. Galton s tables that in 
many cases the whole family had not been recorded, probably the eye-colour of the 
dead or of absentees being unknown. It appeared to me accordingly that it would 
be impossible to deal directly with the problem of fertility. However, it has since 
occurred to me that there is nothing likely to give the missing members of families a 
bias towards one rather than another eye-colour, and that we may simply treat them 
as a purely random subtraction from the total results. Assuming this, Mr. L. N. 
FiLON, M.A., has prepared for me tables of father's and mother s eye-colour and of 
the recorded number of their children. From these* I take first the following 
results, premising (i.) that I call " light eyed," persons with eye-colours 1, 2 and 3, and 
" dark eyed," persons with eye-colours 4, 5, 6, 7, 8, i.e., drawing the line between light 
and dark grey ; (ii.) that I take as small families those with 0, 1, 2, or 3, recorded 
children and us large those with 4 or more recorded children. 



Father. 





Light Eyed. 


Dark Eyed. 


Totals. 


1 


313 


Ul 
139 

280 


1 

454 

403 

1 


264 
577 


Totals 


857 



Mother 





Light Eyed. 


Dark Eyed. 


Totals. 


^ 
M 


253 


202 


455 


& 

3 


225 


169 


394 


Totals 


478 


371 


849 



* Correlation tables were prepared of the size of families to 15, and of the eye-colours 1 to 8, but it 
does not seem needful to print them in ext-enso. 



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MATHEMATICAL CONTMBUTIONS TO THE THEORY OF EVOLUTION. 149 

We have, accordingly, by the method of the present memoir : — 
Correlation of size of family with darkness of eye-colour 

= '0595, for fathers. 
= — -0239, for mothers. 

The former is- just sensible, the latter hardly sensible relative to the probable error. 
So far as they can be relied upon, they would denote that fathers have more children 
the darker eyed they are, and mothers more children the lighter eyed they are. 
This is in accordance with the result given in the memoir, that the modern 
generation is darker than its male and lighter than its female ancestry, but it is not 
the explanation given in the text, although it is probably the true one. If it be the 
true one, dark fathers and light mothers ought to present the most fertile unions, 
and it seemed desirable to test this directly. We have already seen that there 
exists an assortative mating in eye-colour, like tending to mate with like, the 
CO -efficient of correlation being about '1 ; hence if we were to correlate the eye-colour 
of mothers and fathers, i.e., husbands and wives weighted with their fertility, we 
ought to find this result substantially reduced. The following is the table : — 



Fatliers. 



I 



-d 


Light Eyed. 


Dark Eyed. 


Totals. 








<D 








>^ 








P^ 


1183 


612 


1795 


^ 








bO 








>A 








nj 








a> 








,>» 








W 


826 


455 


1281 


^ 








u 








ee 








Q 




1 




Totals 


2009 


1067 


3076 



We find ?• = '0239, or the correlation has been reduced to a fifth of its previous 
value, and is now of the order of its probable error. To mark still further this 
increased fertility of heterogamous unions, I add two further tables, giving the mean 
number of recorded oflfepring for various classifications of parental eye-colour. 



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150 MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 
Fathers. Fathers. 



I 





Colours 
1-3. 


Colours 
4-8. 


Average 
of totals. 


if 

3" 


3-83 


4-57 


407 


§ 00 


3-82 


3-73 


3-79 


Average 
of totals 


3-83 


417 


3-94 



^ 





Colours 
1-2. 


Colours 
3-8. 


Average 
of totals 

3-86 


3" 


319 


4-52 


P 


3-98 


3-96 


3-97 


Average 
of totals 


3-68 


409 


3-94 



The first table entirely confirms all the conclusions reached, — dark fathers and 
light mothers are most fertile absolutely and in union. The second table shows 
that it is the blue-green and grey rather than pure blue-eyed mothers who are most 
fertile. This supplementary investigation accordingly seems to support the view of 
the text of the memoir, namely that reproductive selection is the source of the 
secular change in eye-colour noted, only the prepotent fertility which is replacing the 
blue-eyed element is in the first place that of the dark-eyed male, and only in the 
second place due to mothers having eye-colours dark or light other than true blue. 
We seem accordingly in eye-colour to have reproductive selection working through 
heterogamy rather than through homogamy as in the case of stature.* The effect, 
however, is like, — the progressive elimination of one type of character.] 

♦ See * Roy. Soc. Proc.,' vol. 66, p. 30, and vol. 66, p. 316 et seq. 



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[ 151 ] 



IV. 071 Simultaneous Partial Differential Equations. 

By A. C. Dixon, Sc.D. 

Communicated by J. W. L. Glaisher, Sc.D. 

Eeceived May 9— Read June 15, 1899. 

Contents. 

Pages 

§ 1. Introductory 151 

§§ 2 — 9. On " bidifferentials," or the elements of double integrals, and on the conditions to be 
satisfied in order that a given bidifferential expression may be a complete bidiffer- 
ential 152—159 

§§ 10—13. Theory of equations linear in the Jacobians of two unknown functions; their solution 

reduced to the formation of complete bidifferentials 159 — 162 

^ 14 — 30. Theory of other simultaneous partial differential equations in two independent and 
two dependent variables. A method of solution, with examples of its application. 
One pair of variables is said to be a " bifunction " of other pairs when its bidifferen- 
tial can be linearly expressed in terms of theirs : this idea is of importance in con- 
nection with the derivation of all possible solutions when complete primitives are 
known. Construction of bif unctions in some cases 162—181 

^31 — 42. Differential equations of the second order with one dependent and two independent 

variables. A method of sohition, with examples 181 — 191 

§ 1. In this paper, without touching on the question of the existence of integrals 
of systems of simultaneous partial diflferential equations, I have given a method by 
which the problem of finding their complete primitives may be attacked. 

The cases discussed are two : that of a pair of equations of the first order in two 
dependent and two independent variables, and that of a single equation of the second 
order, with one dependent and two independent variables. 

I follow, as far as possible, the analogy of the method of Lagrange and Charpit, 
and with this object introduce the conception of the " bidifierential " or differential 
element of the second order, which bears the same relation to a Jacobian taken with 
respect to two independent variables as a differential does to a differential coefficient. 

The solutions considered are, in general, complete primitives, that is, such as contain 
arbitrary constants in such number that the result of their elimination is the system 
of equations proposed for solution. The existence of such primitives is sufficiently 
established (see the papers of Frau von Kowalevsky and Professor Konigsberger, 
quoted hereafler) ; it will therefore be assumed, and the object of the investigation 

5.11.1900 



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152 MR. A. C. DIXON ON SIMULTANEOUS 

will be to find conditions that must be satisfied by the equations of the solution and 
to put these conditions in a convenient form for solution by inspection. 

I should add that I am greatly indebted to the referees for their suggestions and 
for help in removing obscurities. 

To the list of authorities given by Dr. Forsyth (' Theory of Differential Equations/ 
Part I., pp. 299, 331), may be added the following : — 

Julius Konig. Math. Annalen, vol. 23, pp. 520, 521. 

Leo Konigsberger. Crelle, vol. 109, pp. 261-340.* Math. Annalen, vol. 41, 

pp. 260-285.t Math. Annalen, vol. 44, pp. 17-40. 
Ed. v. Weber. Mlinchen Ber., vol. 25, 423-442. 
J. M'CowAN. Edinb. Math. Soc. Proc, vol. 10, 63-70. 
Hamburger. Crelle, vol. 110, pp. 158-176. 
C. BouRLET. Annales de I'ficole Normale (3), vol. 8. 
RiQUiER. Comptes Rendus, vols. 114, 116, 119. Annales de Tficole Normale 

(3), vol. 10. 
Lloyd Tanner. Proc. Lond. Math. Soc, vols. 7-11. 
J. Brill. Quarterly Journal of Math., vol. 30, pp. 221-242. 

Several of the above papers are only known to me through abstracts. 

On Bidifferentials, 

§ 2. The idea of a ** complete differential" plays an important part in the theory 
of differential equations. In this paper I shall try to show the importance of an 
extension of the same idea to differential elements of higher orders, such as enter 
into multiple integrals. 

An expression Xc/x + Ydy is called a complete differential when X, Y are functions 
of the independent variables x, y, such that 

8Y/ax = ax/ay. 

If this is the case, then, under certain restrictions, the value of J(Xc/a: + Yo??/) depends 
only on the limiting values of the variables, and not on the intermediate ones by 
which these limits are connected, or, as generally expressed, on the path along which 
the integral is taken. 

This depends on the theorem that 

fdY 8X\ 



\{Xdx + Ydy) = \\(^i-^yxdy 



* For reasons stated below, I am not in agreement with the results given in the latter part of this 
paper. 

t In this paper it should be noticed that the equations (52) on p. 266 are not more general than (46). 



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PARTIAL DIFFERENTIAL EQUATIONS. 153 

when the single integral is taken round the boundary of the area over which the 
double integral is to extend. 

Further, X, Y are in this case the partial derivatives of a single function. 

§ 3. Let us consider the double integral 

\\(Xdydz + Ydzdx + Zdxdy), 

where X, Y, Z are functions of the independent variables or, y, z. It is known 
that this, taken over a closed surface under certain restrictions, is equal to the triple 
integral 

J J J(aX/aa; + a Y/ay + dZ/dz) dx dy dz 

taken over the space enclosed by that surface. 

Hence, if dX/dx + SY/dy + dZ/dz = identically the double integral taken over 
a closed surface vanishes, and taken over two open surfaces with the same boundary 
has the same value ; that is to say, the value of the double integral depends on the 
values of a;, y, z at the boundary only, and not, under certain restrictions, on the 
form of the surface enclosed by the boimdary. 

By analogy we may call the element of the double integral a " complete double 
diflTerential," or a " complete bidiflferential " \mder these circumstances ; the condition 
that X dy dz + Y dzdx + Zdx dy may be a complete bidiflferential is thus 

dX/dx + dY/dy + dZ/dz = 0. 

§ 4. A complete bidiflferential may be expressed as a single term, such as dti dv. 
For let u^ vhe two independent solutions of the equation 

SO that n = a, v = 6 are integrals of the system 

cte/X = dy/Y = d2/Z; 

then ' X = ^|^, Y = ^|^, Z = ^|^>, 

3(y,«)' o(z,x)' d(x,y' 

6 being some multiplier, 

d ax 8Y az_ 3(g,tt,t>) 

3lK dy dz d{x, y, z)' 

Since the last vanishes identically ^ is a function of u, v only ; a function w of tt, v 
may be foimd, such that Zwjdu = 6, and thus 



X 

VOL. CXCV. — A. 



__ 9(w, v) y _ 9(w, v) „ _ d(w, v) 
-d(y,z)' '~8(2,r)' ^-a(r.y)- 



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154 



MR. A. C. DIXON ON SIMULTANEOUS 



Now in finding the value of the double integral taken over a part of any surface, 
it will be natural to suppose the co-ordinates of any point of such a surfia.ce to be 
fiinctions of two parameters, say p, q, and to transform the integral into one taken 
with respect to these. The integral as transformed is 



\\{- 



9(y.2) 



+ Y 



3(g, x), ,j^(x,y) 



+ Z 



3(f2)^ a(2>,j)^ d(p, 



nW"^- 



and the known values in terms of p, q are to be substituted for x, y, z and their 
derivatives, 

ITie subject of integration is 

dw ^t dio dy dw dz hv dx dv dy^ dvdz 

3r 9p 8y 9p 92; 9p ' 9iB 2jp dy dp dz dp 

dw dx dw dy^ dw dz Sv^dx dv dy dv ^ 

dx dq dy dq dz dq* dxbqdydqdzdq 



or 



The integral is therefore 



d(Wt v) 



and if we take a single element we may write 

Xdydz + Ydzdx + Zdxdy=^ dw cZv, 

dropping the parameters jp, q^ since the values which x, y, z have in terms of them 
are immaterial. 

This equation is meaningless unless the expression in terms of parameters is under- 
stood. The same is true of ordinary differentials. If when w is a function of sc, y, z 
we write 

da^^dx + ^dy + ^dz, 

we mean that if x, y, z are supposed to be any functions whatever of a single 

parameter jp, then 

die du dx du dy ^^du^ dz^ 

dp '^ dx dp dy dp dz dp' 

This equation being true quite independently of the expressions assumed for x,y,zm 
terms of jp, we drop the denominator dp for convenience ; but in modem works on the 
Differential Calculus it is quite understood that a differential by itself is meaningless 
apai't fi'om this or some equivalent convention. 



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PARTIAL DIFFERENTIAL EQUATIONS. 155 

§ 5. The fiinctions w, v are not uniquely determined. They may be replaced by 
W, V, where W, V are functions of w, v, one of which, say W, is arbitrary, while V is 
only restricted by the condition 

d(w, v) 

The transformations of ?/;, v which are allowable will thus form a group. For a 
single integral the operations of the corresponding group consist in the addition of 
different constants, that is, in varying the constant of integration ; the theory of 
periodic functions is connected with discontinuous sub-groups of this. It is possible 
that an investigation of the discontinuous sub-groups of the group of transformations 
of two variables which leaves their bidifferential unchanged may lead to an extended 
theory of periodic functions of the two variables. 

§ 6. The finding of the functions w, v may be considered as the indefinite integration 
of the bidifferential expression. It is simplified by Jacobi's theory of the last 
multiplier, which is here a constant. 

we have Xdy — Ydx = ;^ ^^^ "" ^ ^^ > 

and thus, on the supposition that v is constant, 

, Xdv-Ydx Ydz-Zdy Zdx-Xdz 

dw = — ^ = 5 ^ = 5 

ov cv ov 

dz d-x xy 

_ (fi Z - vY)dx + (i/X -^ \Z)dy + (XY -- /xX)rfg 

Hence w may be found, if v is known, by integrating this last expression on the 
supposition that v is constant ; X, /x, i/ may have any values and the constant of 
integration is to be replaced by an arbitrary function of v. Thus, when one of the 
functions w^ v is known, the other is found by ordinary integration. The only 
restriction on the one found first is the equation 

§ 7. Let us now suppose a greater number of independent variables. Let ?^ be a 
function of aj^, Xg . . . o?^. 
We have the relation 

X 2 



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156 MR. A. C. DIXON ON SIMULTANEOUS 

Here the ditterentials represent simultaneous infinitesimal increments, those of the 
independent variables being arbitrary. The equation may also be interpreted by 
supposing «!, X2 . . . x„ to depend in any manner on a single parameter p, when the 
equation 

du * 3m dxr 

^'P ""r=i3'V dp 

holds whatever functions of the parameter we suppose x^ . . . aj^ to be. 

To get the idea of a double differential we must suppose two sets of simultaneous 
infinitesimal increments ; denote them by d, 8. The bidifferential of a;, y is then 
dx . 8y -— 8x . dy* This vanishes if a?, y are not functionally independent, just as dx 
vanishes if re is a constant. The analogy is very clearly shown if we say that dx 
vanishes when some function i^{x) vanishes, dx dy vanishes when some function 
^(Xy y) vanishes. 

If Uy V are functions of n independent variables x^^ x^ . . . aj», we have 

dw = S 5- dXr, 8w = 2 5- SaJr 

** ov * 3t? 

di; = S ^ dXry 8t; = S ^- 80:^, and hence 

du.hv — Zu.do — % S ^ ^ {dXr . hXf — dx^ . 8a:^), 



or 



dudv = X ^ ' dxr dx. 



the summation being taken over all pairs of different suffixes r, s. Hence the 
expression for du dv is formed by multiplying together 

2 5— dXr and 2 ^ cte;. 

with the conventions 

dxdy = — dy dx, 
dx dx = 0. 

We shall often use the notation d{x, y) for dx dy. 

§ 8. For the purpose of double integration of such an expression as 2 X„ d{xr, a?,), 

in which the coefficients X are ftinctions of acj . . . a:„, it is natural to suppose a:^ , . . a;» 
expressed throughout the range of the integration in terms of two parameters, say 
jp, q. The integral thus becomes 

* The dot is used here and throughout the paragraph to distinguish multiplication in the ordinary 
algebraic sense from multiplication according to the Grassmann conventions stated at the end of the 
paragraph. 



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PAETIAL DIFFERENTIAL EQUATIONS. 157 



w 






If X„ = ^ * \ for all pairs of suffixes, the subject of integration in the last 

integral is d{tf^ v)/d{pf q)^ so that the. integral becomes Hdudv. Its value will 
therefore only depend on the values of w, v, that is of jc,, a;^ . . . x^, at the boundary 
of the range of integration, and not on the form of the relations giving 0:^, x^ . . . 
in terms of ^, q^ which define the particular siu'face over which the integral is taken. 
In this case we may write 

2 Kr, d{Xry X,) = d(u, v) 

and call it a complete bidifferential. It is easily seen that the coefficients X satisfy 

the relations 

X,,X(,. + XhX,, + X^.X« = Oi ...... . (1) 

.' -t+t+t-" -f^'' 

for aU combinations of suffixes, where it is understood that the term Xr,c?(a:^, x,) may 
be also written X^d{x,^ Xr), so that 

Xf , = — X,.,. 

The conditions (2) are those which must be satisfied in order that the value of the 
double integral may depend only on the boundary. The difierence of two values 
of the double integral, for which the same boundary is assumed, will be its value over 
a closed surface passing through the boundary curve, and this may be transformed 
into the triple integral 



fffi,(t + l: + th^-^^- 



taken through the voluriie of any solid bounded by this closed surface. Hence this 
integral must vanish for any solid. By taking an infinitesimal solid, for every point 
of which all but a;,-, Xr, Xg are constant, we find the condition (2). 

The conditions (2) would be satisfied by an expression which was the sum of two 
or more complete bidifferentials, but (1) in general would not. 

§ 9. We next try to find whether these c6nditions are sufficient as well as 
necessary. Now all the coefficients X cannot vanish. Suppose that X^g does not, 
then we have firom (1) 

and in virtue of these all the conditions (1) are satisfied. 



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158 MR. A. C. DIXON ON SIMULTANEOUS 

Taking the values thus given for Xr„ X,>, X«- we have 

~ x,jl aPi . ■*" eb-,/ ■•■ XijV &•.• ■•■ -ar,/ "^ x„V a^, "^ a^. / 



Y 



/^ J. ^^ J. ^'Y^ J- ?^\ 4- ^?t/^' -U ^^ 



x„ axjg Xfr axjg Xjj axjg 
X]2 a^'i Xjg aa7j Xj3 atv 

= 1^ (IW) + ^ (r2i) + 1^' (2st) + I" (sit) + |y («2r) + ^ (1^) 

-^13 -*^12 ^12 ^12 -^12 -^12 



M2*^2 



+ 1^ (21«) + ^ (21r) + 1^ (21t). 

Ajj Aj, -a.12 



Thus the conditions (2) are not independent, but all follow from those in which at 
least one of the suffixes 1, 2 enters. If they are satisfied then the equations 

H n 

S Xi^ dXr == 0, 2 Xgy dXr = 

rss2 r=l 

can be satisfied by two integrals of the form w = a, v = h; that is, these last 
equations will give Xj, ic^ as functions of the rest, such that 

&i Xg^ ^ Xrt 

For the conditions necessary and sufficient* for this are the vanishing of such 
expressions as 

J^_ ?!l 4. ?«? . ^_ ?!1J I ^1 ^ ^1 _ 9 Xrt ^ Xe^ j^ Xrt ^ Xyi 9 Xrt 
d^, Xjg Xjj &i Xi2 Xjg OTg Xjj 9.ZV Xj2 Xj3 Oit'i Xij Xjj ocj X^ 

in which 1, 2 may be interchanged and i% s are any two of the other suffixes. This 
expression may be written 

i;, <^l*) " §; (^2s) + || (12r) + ^ ^ (Xi2 X„ + X,, X^ + X^ X,.), 
SO that it vanishes and the conditions of integrability of the equations XXj^Xr = 0, 

r 

l,X2rdXr = are satisfied. If w = a, v = 6 are the integrals, then, since tK^dxr 

r 

does not contain the diflferential of x^, we must have 

* For proof of this statement see Fobsyth, * Theory,* part I., pp. 43-61. 



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PARTIAL DIFFERENTIAL EQUATIONS. 159 



d(u, v) 



and Xir = 



3(«i, av)' 



In like manner Xgr = 6 ^-7 — :, the multiplier being the same. 

Hence ■- ^ = *^! 

Since this vanishes for all combinations of suffixes, ^ is a fimction of u, v, and if 
another function of them, ti;, is so chosen that 

we shall have t X^, d{Xr^ x,) = d{Uy v) = d{w, v). 



r,« 



Linear Differential Equations. 
§ 10. If t^ = a is an integral of the linear partial differential equation 

where X^, X^ . . . X^^.! are ftmctions of x^y . . . x^^^i^ then n satisfies the condition 

S X,^ = 0, 

r=l OXr 

and the complete differential du is a linear combination''^ of the determinants 

dxiy dx^i dx^ • • . dx^y dXf^+i 

^l9 -^> -^ • • • -^J -^+1 

the coefficients in the combination being usually functions of x^, . . . Xm^i. 
If u = a is a conunon solution of the above equation and of 

^1 ai^i + • • • +^ Stew --^-+^' 

then, in like manner, duiaa, linear combination of the determinants 
* This is generally expressed by saying that *' it = a is an integral of the equations 

Xi X2 Xn+i 

For the sake of the analogy with the work of § 11, I prefer the phrase in the text, which expresses no 
more and no less than the one generally used. 



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160 ME. A. C. DIXON ON SIMULTANEOUS 

dxu ^^2> • • • r^n+u 

-^l> ^> • • • J -X-n+i, 

X' Y ' Y' 

but in general, of course, it will not be possible to combine them so as to form a 
perfect differential. 

§ 11. An analogous process of integration may be given for two simultaneous 
equations 

2{Aj,(My - Pjqi)} + SB.?>.- + tC-qi + E = 01 

t{A:^{piqj-pjqi)} + tB\'Pi + tC',qi + -E'^0} * * * * (3)' 

in which the coefficients A, B, C, E, A', B', C, E' axe functions of n independent 
variables, ajj, x^ . . . x^y and two dependent y, 2, and 

To fix the ideas, take n = 3 and let x^ x^ stand for y, z respectively, A;4 for C„ 
Aj5 for — Bj, A45 for E, and make similar changes in the accented letters. Then, if 
tc =z a^ V = h are two equations constituting a solution,* a, h being arbitrary con- 
stants, we must have 



IJ = 1,2..8 



'd(a.bXj) 






(4); 



and the values thus given for p^^ g^, j?2> Q'sj i^8> Q's ^aust satisfy the equations (3) identi- 
cally, since a, & are supposed arbitrary. The equations to be solved are thus reduced 
to others which are linear and homogeneous in the Jacobians, and which do not 
contain the dependent variables. 

The equations (4) give two of the Jacobians of a, v linearly in terms of the 
others ; if we substitute tor these two in the identity 



'*(''.'')=a^)'^^-A 



we find that d{u, v) is a linear combination of the determinants of the matrix of tent 
columns. 

* This solution will not be a complete primitive unless a certain number of other arbitrary constants are 
involved as well as a, 6, a supposition which is neither made nor excluded. 

It may be well to point out that the solution here assumed consists of two equations, and not of one 
equation involving an arbitrary function ; in fact, any solution whatever necessarily consists of two 
equations, and one point of the present method is that these are to be sought together, not successively. 

t For n independent variables the number of columns in the matrix will be |(n + 1) (n + 2), the 
number of rows being still thiee. 



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PARTIAL DIFFERENTIAL EQUATIONS. 161 

d{x^, x^), d{x^, a,), . . . d{xi, Xj) . . , d{x^, x^) 

A' A' A' A' 

There are thus eight bidiflferential expressions, and the problem is to be solved by 
finding such multiples of these as, when added together, will form a complete 
bidifferential. 

§ 12. As in the case of Lagrange's linear equation, this will generally, in practice, 
be done by inspection, and the method will be useful for finding solutions in finite 
terms — when such exist. But in any case,* whether the inspection is successful or 
not, there can be no doubt of the existence of suitable multipliers, in infinite number. 
For it is certain that the equations (3) have — possibly among other solutions — an 
infinity of solutions, each involving two arbitrary constants at least, and any one of 
these may be written t^ = a, v = 6, where a, h are the two constants ; m, v are 
functions of the variables, but may, of course, be implicit functions of great com- 
plexity. The functions w, v must satisfy the conditions (4), and it immediately 
follows that d{u^ v) must be a linear combination of the determinants of the matrix 
formed from (4) as above ; so that a corresponding system of multipliers must exist. 

If the solution is not in finite terms it is not likely to be found by inspection, and 
it is quite probable that the best way to find it would be by solving the original 
equations (3) in series. By whatever means the solution is found, the corresponding 
system of multipliers is thereby determined. 

If nine solutions of the form w = a, v = 6 have been found, the nine 
bidifierentials rf(wi, Vj), ^(ttj* Vg) . . . d{ug^ Vg) must satisfy identically a linear rela- 
tion, since they are all linear combinations of eight expressions only. 

We shall say that one of the nine pairs of functions is a " bifunction " of the other 
eight pairs. 

The following is, then, the definition of a bifunction. When the bidiflferentials of 
any nimiber of pairs of quantities are connected by an identical linear relation, with 
constant or variable coefficients, any one of these pairs is said to be a bifunction of 
the rest. 

The word bifunction is simply used as an abbreviation — at least for the present. I 
am not without hope that at a future time it may be found to have some connotation. 

* If one of the dependent variables with its derivatives is altogether absent from the equations (3), or 
if it can be made to disappear by a change of the other dependent variable, the equations (3) will in 
general have no solution. This case will then be excluded ; it is the only case in which the method of 
solution in series (as given, for instance, by Frau von Kowalevsky, * Crelle/ vol. 80) camiot be used 
to prove that solutions actually exist. 

Another case that may fairly be excluded is that in which all the derivatives of one of the dependent 
variables do not occur or may be made to disappear by a change of the other. Such a system is equiva- 
lent to a single partial differential equation with one dependent variable, since the one whose derivatives 
are absent may be eliminated. 

VOL. CXCV. — A. Y 



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162 MR. A. C. DIXON ON SIMULTANEOUS 

It is, of course, evident that if % v are functions of variables a^i, x^ . . . then the 
pair u, V 18 a. bifunction of all the pairs that can be formed from ajj, x^ , . . Other 
examples will be found later on in the paper. 

§ 13. Sometimes solutions exist for systems of partial differential equations in 
which the number of dependent variables is less than the number of equations. 

If, for instance, with the system just considered we take a third equation of the 
same form, the coefficients being distinguished by two dashes, there may be solutions 
common to the three equations. If u — a^ v = h give such a solution, then it 
follows in like manner that d{u, v) is a linear combination of the determinants of the 
following matrix : — 



d{x^,x,) . . . 


d{xi, xj) . 


Ai2, . . . 


A,y, . . 


A' 

-^ 12> • • • 


A'.. 


A'' 


A.% : . 



Similarly for a greater number of equations. 

Application to other Differential Equations. 

§ 14. There are two classes of equations whose solution depends on that of a pair 
of such linear homogeneous equations as we have just been considering ; they are, 
firstly, systems of two equations in two dependent and two independent variables, 
and, secondly, equations of the second order with one dependent variable and two 
independent. We shall consider them in order. 

Firstly, let y, zhe the dependent variables and .Tj, a;., the independent ; sometimes 
we shall write x^ for y and x^ for z. Let Pi^ p^ be the partial derivatives of y and 
g^i, q^ those of a;, and let the equations be 

/IC^I. ^2. Vy ^y Pu Pzy ?1> Qz) = 0, 

Ai^v ^2. y> 2;, ^i, ^2, g'l, q^) = 0. 

A complete primitive will consist of two equations connecting x^y x^, y, z and 
involving four arbitrary constants. By differentiation these equations yield four 
more involving p^, p^y g'l, g'2- ^^ t^® ^^^o equations are supposed to be a complete 
primitive it must be possible to find expressions for the four arbitrary constants in 
terms of aj^, x^, y, 2, Pi, 5^1, P2> qz I t^® elimination of the four constants must give 
/i = 0,/2=0. 

Let «!, ag, ag, a^ be the constants, and ^i, t^o, w,, u^ the expressions for them in 
terms of x^, Xc^y y, 2, p^, q^y p^^, q.^. Suppose /g, f^yf^.f^ to stand for u^y u^y Vg, %i^ 
respectively. Then by differentiation we have for any value of the suffix i from 
I to 6, 



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PARTIAL DIFFERENTIAL EQUATIONS. 163 

the letter d being used to denote differentiation with respect to x^ or Xo on the sup- 
position that the other is constant, while 3 indicates strictly partial differentiation. 

Since dpjdxx = dpjdx^, dqjdx^ = dqjdx^^ we find by eliminating the deriva- 
tives of jpi, gi, ^2, g^2, that 

J(^i>i>i. 9^1. 5^2) +i>iJ(y.i>i> <iu 92) + qi^^y Ply qu ^2) + J(^2>P2> 9i> 92) 

+ Pr^iVy V^y 9\y 9%) + 92^{^yV2y 9\y 9%) = 0, 
and %!, ji, ^1, p^) + ^1%, ^1, p^, p^) + q^J{z, q^, p^, p^) + J{x^, q^, p^, p^) 

+ P2^yy 92y Ply Pi) + 92^{^y ?2>Pl>i^2) = 

where J( ) denotes the Jacobian of any four of the functions /j, f^yfzyf^^fhyA ^*^^^ 
respect to the variables specified in the bracket. Of these equations there are thirty, 
but since they are given by the elimination of six quantities from twelve equations 
only six of the thirty can be independent. 

§ 15. One pair of these auxiliary equations will contain Jacobians of /i, /g, ^, f^y 
and will in fact express the conditions that the equations 

dy = p^dxi + p^dxc^ 

dz = q^dxi + q^dxc^ 

shall be integrable without restriction when p^, p,^, q^, q^ have the values given by 
the equations /i = =fc^, /g = a^,f^ = a^. 

Thus, if a pair of functions ^, J^ can be found satisfying these two auxiliary 
equations, the solution can be completed by solving a pair of simultaneous ordinary 
equations. (See Mayer's method, Forsyth, * Theory of Differential Equations,' pp. 
59-62.) 

The two auxiliary equations that^^ f^ must satisfy are linear and homogeneous in 
their Jacobians, the coefficients of the Jacobians not involving the functions f^, f^ ; 
the number of independent variables is apparently eight, but it may be taken as six, 
since two of the eight variables x^, Xg, y, z, P\y p^y 9\y 9^ ^^^ given as functions of the 
other six by the relations /j = 0,^ = 0, and may be supposed eliminated from^j,/^, 
if that is desirable. 

The colimins of the matrix formed as at § 11 are the rows of the following array : — 

c?(xi, Xg), 0, 0, 

d{x^y y\ 0, 0, 

d{x^y z\ 0, 0, 

d{xzy y)y 0, 0, 

(5) d{xz, z), 0, 0, 

Y 2 



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164 MR. A. C. DIXON ON SIMULTANEOUS 

%,«), 0, 0, 

^{^u Pi). (91. 93} » {Pi,qi}> 

d{x^,p{), 0, {Pi,q'i], 

%. Pi)>Pi{qv %} pAps, 9i} + PiiPi, qi), 

(10) d{z, Pi),gi{?i, q,], qAPi, qi] + q^iPi, q^], 

(^i^!lyPi)>{q^yqi}> {q2,pi], 

^iy'Pi\Pi{quqi]> PiiquPi) +Pi{qiyPi]* 

d{^>P2)> qdqu qi}y qi{q\>Pi) + qi{qi>P))> 

d{yi,qi),pi{qci,Pi] •\-P2{qi,Pi}, PifPi.i'al. 

c^(2.?i). ^ifo'z.i'i} + qi{qi>Pi}, qi{px.Pi}> 

d{xi,qi),{px,qi}y 0, . 

(20) d{Xi, q^), (pa, q^}, f i>i, P2} , 

%. 92), I'll;?!, qi) + P2(P2. g'l}* i'2{pi.i>2}. 

«^{«. 72), g'lfPl. fZi) + 92(l>2. 9l}. 92{2'|.i>2}. 

^0^1. M 0. {a^i, 9i} +i>i{y. 9i} + 9i{«.9i} + {a'2. 92} +Pi{y> 92} + ?s{2. 92}. 

c?(Pi. 9i). {^\,qi\ +Pi{y, 92} + Q'll^. q%}> -{^i>Pi} -pAy^Pi) - g'j2.i'2}> 

(25) ci(pi,q.2), - {Xi.^ri} -Pi{y,9i}-9i{2.9i}. -i^i^Pa) -PziV^Pi) - 92KP2}. 

%2. 9i). W 92} + P2(y.9a} + qd^^qi), {aJi.Pi} +2>i{y.Pi} + gJ^^Pil. 
^(P2.</2).-{a;2.!?i}-P2{y.'?i}-?2{2.g'i}. {3^2,^1} +i?j{y,i?,] +g2{2,i>i}, 
%i. «?2).{a^i. Pi) +Pi{y>Pi] + 9i{2,pi} + {a'2.i>2} +Pi{yy P25 + 92(2. qi}> 0, 

(5) 
Here {p^, q^}^ for instance, is written for 3(/i,^)/3(Pi,g'i), and eveiy fifth row is 
numbered. 

§ 16. In order, then, to solve the equations /^ = 0,^ = we have to form such a 
linear combination of the determinants of this array as will be a complete bidiffer- 
ential, say <l{f^, f^, f^, f^ being such functions that the equations /^ = :=^fc^^f^ = aj, 
f^ = a^ can be solved for jOj, q^, p^, q^. The array contains twenty-eight rows, but 
thirteen of these are combinations of the other fifteen. For instance, multiply the first 
row by dfjdx^, the second by dfjdj/y the third by S/J/S^, the seventh by 3/i/3pi, the 
eleventh by dfjdp^y the fifteenth by Sfjdqiy the nineteenth by 9/i/9g2 a^d add ; the 
resulting row is 

^(^i/i), 0, 0, 
which vanishes. Other vanishing rows may be formed similarly by combining the 
rows of the array so as to have in the first column one of the following — 

(^i^iJll d{x.^J\)> d{yj^\ d{z,f^\ d{p^Jl)y d{p^yfl)y d{q^Jl)y d{q^yf\)y 
%i>/2)> d{^^2yA\ %>/2)> d{zj^\ dip^yAl d{p^J^\ d{q^J^\ d{q^J^). 



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PARTIAL DIFFERENTIAL EQUATIONS. 165 

The coefficients in these combinations are partial derivatives of f^ or ^, thus, for 
instance, 

d^vM = ^ rf(Pi. ^.) + ^ d{p,, X,) + I rf{^i, y) + I rf(;>„ z) 

and so in other cases. 

The number of these combinations is sixteen, but it is to be lowered by three, 
since d(f^^f^ and d{fci,f^ are identically zero and (^{fi,/^) can be formed by com- 
bining the sixteen in two ways, so that three linear combinations of the sixteen 
bidiflferentials vanish identically. 

Hence the array contains virtually only fifteen rows (28 — 13) and as there are 
three columns, we have thirteen bidifferential expressions to combine. Any pair of 
the four Amctions ajj, ^o, y, z will satisfy the two auxiliary equations, as is clear either 
from the equations themselves or from an examination of the matrix ; of course these 
solutions of the auxiliary equations will not give a complete primitive. 

§ 17. If a complete primitive has been found it leads, as has been explained, to 
four equations 

Ui = «!, Wjj = a^y ^^3 = ag, u^ = a^, 

and any pair of these must satisfy the auxiliary equations. Thus twelve pairs of 
functions satisfying these are known, namely 

a:,anda:^ {ij= 1, 2, 3, 4) 

u, and Uj {i,j = 1, 2, 3, 4). 

'These, however, are not all independent, but one pair is a bifunction of the other 
eleven. 



For if <^(a?i, x^, x^, x^, a^, a^, ag, a^) = 



<^(a?i, X.2, x^, x^, tti, flo, ag, aj = 01 
^{x^, 0^2, ajg, x^y a^, ag, ag, a^) = oJ 



(6) 



are the equations of the complete primitive, they must reduce to identities when 
t/i, U.2, u^y u^ are substituted for a^, ao, ag, a^ respectively. 
Hence 

4>{^ly ^Z. ^3» ^J» '^l' ^2> '^3> ^-^i) — ^ \ /yx 

^{x^, a-a, ajg, x^y Uj, W2, u^y W4) = j 
Identically, and ^ 34> ^ = . ^ -^* du,, 

t^dxi = — 2 >.* diHy 

i oxi i aui 

and the bidifferentials of the twelve pairs of functions are connected by a linear 
relation. 



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166 MR A. 0. DIXON ON SIMULTANEOUS 

§ 18. The method of Charpit for a single partial differential equation of the first 
order shows how all solutions may be deduced from one complete primitive, and it is 
a question of interest and importance whether there is any analogous method for 
simultaneous equations. Now it follows at once from the conditions for a complete 
bidifferential that a bifunction of the pairs that can be formed from m functions, say 
u^, Wg . . . . Wm, will be a pair of functions of u^ . . . . Um- In the present case a 
bifunction of the six pairs that can be formed with u^y t/^, t/g, u^ will be a pair of functions 
of these four, and the complete primitive to which it will lead will be the same as 
that given by t^i, w^. For when a solution of the auxiliary equations is known it leads 
directly to one and only one complete primitive by the integration of the equations* 

^y = Pi ^^1 + Pz ^^2 

dz = q^dx^ + 9^2 ^^2; 
also the complete primitive to which the equations F^ {u^y t/g, u^y u^ = const., 
F2 (ti], t/2, ^^3, u^ = const., will lead can be no other than is given by 

u^ = ai, 1^2 = ^2» ^'3 = ^s» "^4 = ^4- 

It must not, however, be forgotten that the system F^ = const., F2 = const., 
y^ = 0, /2 = may have a singular solution. If F^, F2 involve two other arbitrary 
constants this singular solution will involve four, and therefore in general be a com- 
plete primitive of the equations yj = 0, ^ = 0. Moreover, all new complete primitives 
are included among those thus given. 

For every solution implies six equations connecting x^^ 0^2, y, z^ Piy qi, p^y q% (two 
of these six are of course f^ =: 0, f^ = 0), and, therefore, by elimination of 
^1, ^2y y^ ^> Pi> 9^1 > i^2> ?2> ^^^ equations or more connecting Mj, Uc^^ %, w^, which are 
known in terms of these eight quantities. If Wj, u^^ u^y w^^are connected by four equa- 
tions they are constants, and the solution is therefore included in the old complete 
primitive. Let us, then, suppose that w^, Wg, u^y ^4 are connected by two or by three 
equations, 

F>i, W2, 7^3, u^) = (a = 1, 2 or 1, 2, 3). 

Now if pi, pc^y g^i, g^2> are all expressed in terms of jo, q, two of their number, and 
^i> ^2j 3/> ^y ^y ineans of the equations /^ = 0,y^ = 0, the expressions 

^y "" Vidxi — pc^dxc^y dz — q^dx^ — qc^dxc^ 
must both be expressible in the form 

K^dui + A^du:^ + Agdwg + A^du^y 

* Otherwise thus — ^if in the auxiliary equations we suppose /s to have the known value Ui, they become 
a pair of linear equations for /4, which must be satisfied by u^, Us, U4 ; now two linear equations in six 
independent variables can only have four functionally independent solutions, and one of these is known, 
namely, Wi. (In exceptional cases the two linear equations for Wg, ih, u^ may be equivalent ; for instance, 
suppose fi = pi + qiy Ui = p2 + <tiyh having any form.) Hence, except in special cases, the particular 
complete primitive is defined when one of the "functions %\y t*2, t^, w^, or more generally a combination of 
them, F (Wj, w^? ws, W4) is known. In the case supposed in the text two such combinations are known. 



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PARTIAL DIFFERENTIAL EQUATIONS. 167 

and since dp^ dq are absent we must have in each case 

2 A,v: =0, t A,z- = 0. 

r=I op r=l Oq 

Thus.the equations 

dy = p^dx^ + pcidxo^ dz = q^dx^ + g^^^aj^ 
become 

1 dii^y du^i du^^ du^f 
dui Bi/g du^ du^ 
^' a^' 3^' 3^ = 0. 

3^1 3m2 9^8 du^ 

a^' 3^' a^' 3^ 

These two equations, connecting dui^ dii^^ du^^ du^y taken with the system 

* 3F 

2 ^ dur = (a = 1, 2 or 1, 2, 3), 

show that if Wj, w^, 7^3, U4 satisfy by themselves no other relations than F« = 
(a = 1, 2 or 1, 2, 3) we must have, as a consequence of the equations of the 
solution, 

r=l dlCr 3p ' r=l 3Wr S? 

If, then, there are two equations 

Fi = 0, F2 = 0, 
the four equations 

S|^|'=0.S 1^1^=0 (.= 1.2) 

must reduce to two only. This will be the ordinary case, and we see that if the 
forms of Fi, Fg, have been found by any means, the solution is completed without 
integration ; the process corresponds to Charpit's method of deducing all complete 
primitives from one, but it diflfers in that the functions F^, F^, are not arbitrary ; 
they must, in fact, be so chosen that the four equations last written shall reduce to 
two, and the conditions for this are clearly very complicated in general, though in 
particular cases available forms for Fj, F^ may be seen on inspection. 
In the more uncommon case, when there are three equations 

F, = 0,F„=0,F3=0, 
the six equations 

ii 3;^ V "" ' ii 3z.. "3^ - (a - 1, 2, 3), 

must reduce to one only. 

These two cases are further discussed, from a somewhat diflferent point of view, 
in §§ 21—23. 

It should not be forgotten that the form in which tlie new complete primitive has 



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168 MR. A. C. DIXON ON SIMULTANEOUS 

just appeared is not that in which complete primitives were discussed in § 14, since 
the equations are not here supposed to be solved for the arbitrary constants. 

§ 19. In addition to the six pairs {u,, Uj) of functions satisfying the auxiliary 
equations, we have also the six pairs (oc,-, Xj) ; of these twelve, eleve i are indepen- 
dent, the other being a bifunction of them. If we can find : bifunction of the 
eleven pairs which is not a bifunction of either set of six it Avill give a new complete 
primitive ; whether every, or indeed any, other primitive is thus given is a matter 
for further inquiry. 

Suppose Vi = hi {i = 1, 2, 3, 4) to be a new complete primitive, then it gives six 
more pairs of functions satisfying the auxiliary equations, and thus we have in all 
eighteen pairs. The bidifferentials of these must be connected by (18 — 13) five linear 
relations, one of which has been written (8) ; by means of the other four, an 
expression of either of the following forms — 

Ad(v2, t's) + Bd{v^, Vi) + Crf(t'i, r.), 

can be found which will be equal to a linear combination of the twelve bidifferentials 
d {xiy Xj) and d {uiy uJ). It is natural to ask whether, conversely, any linear combina- 
tion of these twelve which can be written in one of the above forms will lead to a 
complete primitive ? In the first case this is not so, for if we take any fiinction 
whatever, tj, of six independent variables, fj, ^^ . . . fg, we may choose the coefficients 
<*!>••• «6> so that 

shall be a linear combination of eleven* given bidifferentials ; the expression t, oLi d^i 
may then be reduced to three terms, P^ dt,^ + jSg dt^ + ^Sg djg, so that for an 
arbitrary function {t}) a combination of the eleven given bidifferentials can be found 
of the form fi^ d{r)y £]) + ^82 d{yi, t,^ + ^Sg d[yiXz)> which is the same as Ao?(i;i, v^ + 
Brf(t;|, Vg) + Cd(vi, V4). This argument does not apply to the second form 

Ad(t?2, ^^s) + M^3. '^i) + C!c/(vi, v^\ 
and further investigation may show that any combination of the eleven that can be 
reduced to this formf will lead to a primitive. 

* Not of any lower number in general, since the most general bidiiferential expression in this number 
of variables contains fifteen terms, while the expression just written vanishes identically if 

so that there are virtually only five coefficients, of which one must be left arbitrary. 

t The conditions necessary that a bidifferential expression may be reducible to this foim include 
algebraic ones which are the same as for a complete bidifferential, since 



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PARTIAL DIFFERENTIAL EQUATIONS. 169 

§ 20. Before we can claim in any sense to have found the general solution of the 
auxiliary equations, we must be in possession of thirteen pairs of functions satisfying 
them ; we have only eleven when we know one complete primitive, and hence one 
more complete primitive, or even possibly two, must be found. An example (below, 
§ 29) will show that one more is not always enough. 

It is perhaps worth while to remark that any complete primitive defines the 
whole system of solutions, since it defines the differential equations. 

§ 21. The question of finding new solutions when a complete primitive is known 
may be attacked by the method of varying the parameters. Take the equations 
(6) or (7) of § 17. The problem is then to find such variable values for Wj, u^y Wg, u^ 
as will satisfy the equations 

S P^dui = 0, 2 ^^dui = (9). 

Since all variables are supposed functions of aj^, cc^, we may make one of two 
suppositions with respect to w^, t^g, u^, u^ ; either they are connected by three 
relations and are all functions of the same variable, say ty which is ot course a 
function of ar^, Xg, or they are only connected by two relations, so that two of them 
may be taken as functions of the other two. 

Suppose first that they are all functions of the one variable t Then, generally, 
the four equations (7), (9) will define a;^, Xg, ocg, x^ also as functions of ^, and hence 
this supposition is not admissible unless it is possible to choose the functions of t in 
such a way* that the four equations (7), (9) will be only equivalent to three. The 



If these conditions are satisfied by an expression 
it can be put in the form 



2 Ai//(a'i, Xj), 

t,; = 1.2. . 6 



and then it must further be possible to express 

6 8 

2 ki dxi and 2 /i^ dxi 

i=l i=l 

as linear combinations of three differentials, dvi^ dv^y dv^ The discussion of the conditions therefore 

belongs to the theory of the reduction of two such expressions, that is, of the extended Pfaff problem. 

* It seems obvious that this will not generally be possible ; but it may be well to give an example. 

Suppose the complete primitive to be 

y = axi^ + bx2 + c, ^ 

z = cxi + ex2^ + bxiX2^j^ 
so that the differential equations are 

y = ipi^i + p^ + qi - pix^, 

z = qiXi + ij2a"2 ~ P^iX2\ 
then the variations of the pammeters a, &, c, e must satisfy the equations, 

Xi^da + xodb + dc = 
Xidc + x-^de + XiX-^dh = 0; 
VOL. CXCV. — ^A. Z 



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170 ME. A. 0. DIXON ON SIMULTANEOUS 

number of conditions, which will be of the nature of ordinary diflferential equations, 
thus imposed on the four parameters must not be greater than three ; for if they 
are subjected to four conditions they are made invariable ; it may be, however, 
less than three. For instance, a complete primitive of the equations jpi=:p2» 9i = ?2 
is given by 

y = a{x^ + Xg) + 6, 2; = c{x^ + 3^2) + e ; 

the equations given by varying the parameters are 

{x^ + Xc^)da + c?6 = 0, 
(xj + X2)dc + de = Oy 

which give the single differential equation connecting the parameters 

da de = db dc. 

We may then assume arbitrary forms for two parameters in terms of a third, and 
find the fourth by integration. Say, for instance, 

b = <f>{a)y c = V^(a), 
then e = J<^'(tt)i//(a)da, 

a^i + a:^ = — it>\a) ; 

thus we arrive at the known general solution 

y = X(^i + ^2)> ^ = ^(^1 + ^2)- 

whence, by elimination of xi, 

{x^Hh + dcy{x^b + dc) + x^iHe^da = 0. 

This equation must fail to define 0^2, so that 5, c, and a ov e must be constant ; thence it follows that all 
four parameters must be constant. 

I lay stress on this, because it is not in agreement with the results of Professor Konigsberger (' Grelle,' 
vol. 109, p. 318), and appears in fact to show that his method there given is faulty. Professor Konigs- 
berger assumes (p. 313; I take m = 2) that the most general integral of the equations 

f\{^\y a^2, y, z, pi, JP2, qu ^2) = 

Mxu X2, y, z. Ply p2y qi, q^) = 
has the form 

y = wi(ari, iC2, </>i[^i(a^i, iCa)], <h[h{^u ^2)]) 
z = ft>2(ici, X2, </>i[^i(a^i, 0:2)], </>2[^2(a:i, a-^i)]), 

where </>i, <^ denote arbitrary and ^1, ^2 definite functions. But suppose these equations solved for 
<^, <^2 iu the form 

^\\^\{^\y «2)] = Xi(^i» ^h y, ^) 

H^i(xiy »2)] = X2(«i, ^y y, ^) 

and the arbitrary functions eliminated by differentiation. The differential equations thus formed are of 
the first degree in |?i, ^2, S'l, ?2» *^^ *^® ^^^ V ^"7 n^eans of the general form assumed. The differential 
equations in the examples given by Professor Konigsberger are, in fact, linear (see pp. 319, 328). The 
method appears to be founded on an interpretation of the last clause of § 2 (p. 290), which is not justified. 



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PARTIAL DIFFERENTIAL EQUATIONS. I7I 

In the case of two equations of Clairaut's form 

y = Pi^i + Ih^^ + <^(Pi, i>2. 9^1. 92). 
z = q^x^ + q^x^ + y^fip^, p^, q^, q^\ 

which will be more fully considered later, the number of differential relations among 
the parameters is twOy so that one parameter may be taken as an arbitrary function 
of a second, and the other two found in terms of the second by solving two ordinary 
differential equations. 
If the primitive* is 

y = aa + 6)8 + cy + ^8 
^ = Aa + B^ + Cy + E8, 

where a, 6, c, e are the parameters. A, B, C, E known functions of a, 6, c, c, and 

^y Py y> 8 known functions of x^, Xc^, then the variations of the parameters must 

satisfy the relations 

cda + fidb + ydc + Sde = 0, 

cudA + fidB + ydC + SrfE = 0, 

and thus, in general, if a, &, c, e are all functions of one variable they are connected 

by three relations 

dA/da = dB/db = dC/dc = cflE/de. 

The integral equivalent of these equations consists of three relations connecting 
a, 6, c, e with three arbitrary constants, and by eliminating a, 6, c, e we find a new 
solution of the original differential equations which is not a complete primitive, 
since it only contains three arbitrary constants. 

These examples show that the number of conditions to be fulfilled by the para- 
meters when all four are taken to be functions of one of them, may be one, two, or 
three ; this number is to be made up to three by assuming arbitrary relations (two, 
one, or none, as the case may be). 

§ 22. Usually the parameters will not be functions of one variable only, and we 
may suppose two of them, Wg, u^, to be functions of the other two, w^, u^. 

The partial differential coefficients 

du^ du^ du^ du^ 
dill * du^ ' du^ ' dtt^ 

are then given by the equations (9), each of which is equivalent to two. The first, 

for instance, gives 

3<^ 90 duj^ dcf) du^ 

dui dic^du^ dti^^dtii ' 

3^ 9^ du^ d<f> du^ 

9^2 du^du^ dic^du^ 

The derivatives are thus given in terms of le^, w^, Wg, u^, iCj, x^y ajg, x^, and the last 

* It is imnecessary to give the differential equations. 
z 2 



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172 



MR A. C. DIXON ON SIMULTANEOUS 



four may be eliminated by means of the relations (7) ; so that in the end we shall 
have two relations connecting w^, u^^ Uo, u^, and the derivatives ; the problem is of 
the same form as the original one, to solve two simultaneous partial differential 
equations in two dependent and two independent variables. 

Interchange of Variables and Parameters. 

§ 23. A curious thing may be noticed at this point. If in the equations <^ = 0, 
^ = 0, we treat x^, x^^ Xz^ x^ as arbitrary constants and eliminate them by differentia- 
tion, we are led to the same differential equations connecting w^, Wg, W3, u^ as were 
just now given by the variation of parameters. Thus two equations in two sets of 
four quantities will give two pairs of simultaneous partial differential equations by 
taking each set of the quantities in turn as variables and the other as arbitrary con- 
stants. The auxiliary equations, if expressed in terms of the eight quantities, will 
be the same in both cases ; this gives a meaning to the six solutions of the form 
(x,-, Xj) which we found the auxiliary equations to have, for any one of the six 
wiU lead to the primitive <^ = 0, ^ = of the second pair of differential equations, 
just as a solution {ui, uj) leads to this primitive for the first pair ; any new solution 
of the auxiliary equations will in general lead to a new complete primitive for either 
pair, but an exception to this rule will arise when, for instance, the x differential 
equations have a complete primitive which gives three relations among u^, u^^ W3, m^. 

The array (5), transformed so that the variables are aj^, x^, ajg, x^, u^y Uc^, M3, u^, 
connected by the equations ^ = 0, t/r = 0, will have six rows of the form 

d{xi, xj), 0, 0, 
six of the form ^(w/, w^), 0, 0, 

and in the other sixteen there will be 

d{xi, uj) in the first column, 

in the second the minor of ^ ^ in the determinant : 



^<f> 


d^<f> 


S?<f> 


3^^ 


30 


dyjr 


ctejBMi ' 


d^idu^ * 


ari8?tj ' 


dxjdu^ 


arv 


a.-i 


d'-4> 


d';f> 


d^<f> 


d'4, 


d<t> 


3^ 


dx^dic^ ' 


d-Xc^du^ ' 


dx^ii^ ' 


djc^ii ' 


Br,' 


3iCjj 


a««^ 


av 


^<f> 


3'*^ 


d<f> 


Byfr 


ar>i' 


a^3.^' 


a«s3?*3 ' 


a^^jSw,' 


3.3' 


dx^ 


d^4> 


d'if. 


3*0 


5^ 


d<f> 


d^lt 


dx^Ui ' 


dx^d^Uj ' 


dx^duj' 


dx^du^ ' 


dx,' 


^x^ 


d<f> 


dtf, 

9^2 ' 


Btf, 
9«s' 


3<^ 
3«/ 


0, 









3«3' 


Bylr 


0, 






(10) 



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PARTIAL DIFFERENTIAL EQUATIONS. I73 

in the third the same expression with <f>, \p interchanged. The array is thus practi- 
cally unchanged by interchanging the sets x and w, as should be the case. 
§ 24. This transformation may be accomplished by taking the equations 



from which may be deduced 






d_ I d^ du{\ __ ^ /^ 9<^ dUi 
dx^ \ i dui dxj dx^ \ i bui d(x^ 

^^duiduj ^ff> duj d^<f} duj . d^ dui 

i j duidi(j dx^ dx^ i duidx-i dvc^ i duidy -^^ dx^ i duidz ^^ dx^ 

— ^^ ^^ ^M»rf^^j , ^ 9^<^ dUj d^4> duj 9^^ rfwi 

i j dui^Uj dx^ dx^ i duid.Vcj^ dx^ i duidy ^^ dx^ i dicidz ^^ dv^ ' 

Now Pi, q^y p^y q.^ are given by the relations 
and hence this equation may be wiitten 



* \dx^ d{x^,y,z) J i \dx^ d{x^,y,z) J 



(11); 



in this if), xff may be interchanged so as to give another equation. 

Now, suppose ^ = aj, X = % to be two of the four equations connecting 
^1, ^^2j ^3? ^4 with ccj, x^y which yield a new complete primitive, and that y, z have 
been eliminated from ^, x by means of the equations <^ = 0, ^ = 0, then the deriva- 

*^^^® dx* dx' ^^'^ ^^ given by the following relations : — 

^ 9<^ dUi _^ Q 
i diti dxi ' 

^d±dui _ ^ 
i 9m; dxj 

.d0 dUi , 00 
i oiii dx^ co\ 

i dui dj\ D./?! ' 

and similarly for the derivatives with respect to x^. 

Substituting the values hence found for these derivatives in the equation (11), we 
have an equation linear in the Jacobians of the form 



l^''y(i=l,2;J=l,2.S,4). 



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174 MK. A. C. DIXON ON SIMULTANEOUS 

the coefficient of the Jacobian written being the minor of g^rg^ ^^ ^^^ determinant 

(10). Hence the constituents in the second column of the transformed array are as 
stated, and those of the third are found in like manner. It is not, of course, neces- 
sary that these columns should be the same as would be found by actual substitution 
of the values of pi, jo^, Ji, q^ in the columns of the original array ; a linear trans- 
formation is allowable, with constant or variable coefficients. 

The above process gives fifteen independent rows of the array ; the others are 
deduced from the consideration that y, z are known in terms of x^^ x^^ t/j, tfcg, u^^ u^ 
from the equations ^ = 0, ^ = 0. 



Examples. 

§ 25. I. As a first example of the method of solution, take the equations 

ttj = ttg, ^1 = ^3, 

where aj, ^S^ denote known functions of ajj, jp^, q^ and ag, fi^ known functions of 
a^2> P2> ?2- 

In the array (5) multiply the seventh row by ^~^^y the fifteenth by g;^^' ^ L the 

twenty-fourth by - ^^^ ^l , and add. The result in the first column is d{aL^y fii), in the 
second, by virtue of the particular forms of/^ and/gj 

{^i.i^i} {?!> ?2} + K. 9i} {Q2rPi] + {Pi> ?i} {a^i» ^2} ^r 0, 
and in the third, 

i^i^Pi] {^2. 9^1} + {^1. qi) {PiyP2] - [Pu qi] {^1,^2} or 0. 

Hence a^, ^Sj are two functions satisfying the auxiliary equations, and a solution is 
given by finding p,, q,, p^, q^ from the equations 

^1 = «2 = ^> 
^1 = ^2 = &> 

and integrating. Two constants will be introduced by integration, so that the result 
is a complete primitive. 

§ 26. II. Take, secondly, the equations 

y = Pi^i + F (0:2, i>i, g^i, i>o, q^\ 
z = q^x^ + G {x^, ^1, 5i, 7)2, q^y 
Here the twenty-fourth row is 

^bi> ?])> 0, 0, 



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PARTIAL DIFFERENTIAL EQUATIONS. 175 

so that pi, 9i are two functions satisfying the auxiliary equations, and the integral is 
to be foimd by putting p^ = a, 5, = h. Thus we have 

y — axi = ¥ (Xz, a, h, p^, q^), 
z~bXi = G (x^, a, h, p^, q^), 

or 7i = F{la,h,v',i% 

C = G{i,a,b,y,',C), 
where ^ = cCg, 7^ = y — ax^^ ^= z — hx^^ 

r,' = d7,/d^, i' = dildl 

These are ordinary differential equations, the solution of which will involve two 
new arbitrary constants and so constitute a complete primitive of the original 
equations. 

§ 27. HI. The equations 

y = l^i^i + P^^ + <^( Ply P2> ?i. <1^\ 
z = q^x^ + q^x^ + V^(^i, Pa, q^, q^\ 

are of special interest, because more complete primitives than one can be found. The 
obvious solution is p^ = a^, p^ = ag, qi = 61, q^ = &2> 

y = a^Xj + a^Xc^ + <^(ai, ag, &i, 62)^ 
z = b^x^ + h^x^ + y^{a^, %, 61, h^. 

Suppose a^ a^^ h^y h^ to be variable, but functions of one variable only— say aj, then 
their variations must satisfy the relations 

x^ da^ + x^ da^ + ^<^ = 0, 
Xy dh^ + x^ d\ + ^'A = 0. 

These define x^, x^, and, therefore, also y, 2 as functions of a^, unless the determi- 
nants of the matrix 

da^y da^y d(f> 11 

db^y db^y d^ I 

vanish ; it is necessary, then, that these determinants should vanish. Thus 
^u &i» ^2» ^2 ^re connected by two ordinary differential equations. We may assume 
any third relation connecting them at will ; suppose b^ = ^{^1)9 ^ denoting an 
arbitrary function. Then by integration we may suppose a^, &2 fo^^d in terms of a^. 
Also a^ is connected with x^y x^ by the relation 

^^ + ^2 da, + da, "" "* 

SO that ttj, 61, a2, 62 ^^e all known in terms of Xj, oTo, and by substitution the values 
of v, z are found. 



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so that 



176 MR. A. C. DIXON ON SIMULTANEOUS 

§ 28. The solution may be verified. We have taken h^ = F(ai), a known but 
arbitrary function of a^, and a^, h^ other functions of aj, such that 

Then we have the further relations 

which are of course not distinct. Also 

P^-^^^ de, [^ ^^ ^2 rfoj ^^ 3a, ^^ 3^3^01 ^ 36i da^ ^ 36, (iaj " ^^' 
and in like manner p^ = a^. 

Again 2? = 6i iTi + &2 ^2 + ^{<^iy %» ^i» ^2)^ and 

^ «- 7, I ^ L ^*l I ^ rf^g 3^^ 32fr da2 3i/^ 

9^1 - oi i- ^^ i^a^i ^^^ "<■ ^2 ^^ -h 3^^ i- 9^^ ^^^ "^ 36^ rfo, "^ (^6. daj "" ""^^ 
and similarly g^ = 6^. 

Hence the original differential equations are actually satisfied. If the arbitrary 
relation assumed — which may if convenient involve more than two of the parameters — 
contains two arbitrary constants, the new solution will generally be a complete 
primitive, since two more constants are introduced by integration. *t 

* The ordinary equations to be integrated may have a singular solution with one arbitrary constant, 
or with none : if the arbitrary function has been chosen so as to involve three or four arbitrary constants, 
the whole number being thus raised to four, the solutipn so given may quite well be a complete primitive, 
and, in general, will be so. 

t The above investigation in a modified form shows how to find integrals of a system of three 
equations 

/2 (w, J, i?i, ;?2, gi, ?2) = 0, L (12) 

/s (w, t;, i>i, i>2, gi, ^2) = o,J 

where u = piXi + p^x^ - y,v = qiXi + q^x^ - z. 

One solution is to take t^)?M'i,^2)^ii $2 £U3 constants connected by the three relations (12) ; if they are 

not constants we have 

du =■ Xi dpi + iCa dp^^ 

dv = xi dqi + X2 dq2. 



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PARTIAL DIFFEllENTIAL EQUATIONS. 



177 



§ 29. Let us now consider the new solutions of the auxiliary equations, given by 
the new complete primitive. The old solutions are the six pairs of the form a;,-, xj and 
the six of the form t<,-, tij, where Mj = pi, % = p,, Wj = g-j, tt^ = q^. The bi- 
differentials of these twelve satisfy the relation 

d{y, z) -pAx^, z) -pod{x^, z) - q^d^, x^) - q^y, x^) + {p^q^ - Piqi)d{x^,Xi) 



+ 



x^ 




_^3^ 






3f 


«1 




3* 
3j, 




3^ 




«2 




3<^ 

3?8 




3^ 


3^ 



^'(Pi.l'a) + 



<^(Pi. 92) + 



«?(;>2. g'2) + 



X, 


.3* 


3<^ 




3f 




«1 


3^ 


3<^ 
33i 




3^^ 


3^ 
+ 3j. 




3« 


3(^ 




a?.' 


^ 


Xl 




3Vr 



^(Pi> 9i) 



^(i^2> 9l) 



<^(<?l» ^2) 



In the auxiliary equations we may take ajj, x^^ pj, p^^ q^y q^ as independent variables, 
since y, « are given explicitly in terms of these six. 

From (12) follow three more relations connecting the six differentials du, dv, dpi, dp2, dqi, dq2, so that 
their ratios are determinate, and therefore w, r,^i, qi,p2, Ja can only be functions of one variable. The two 
equations last written will then, generally, give xi, x^ in terms of this variable, which may not be. Hence 
we must have 

du = Xdv^dpi s= \dq\^dp2 = ^^2j 

and since df\ = 0, c(^2 = 0, dfz = 0, and du^ dvy dpi, dqi, dp^, dq^ do not vanish, X must satisfy the 
equation : 

Sib dv ' dpi dqi ' dp2 dq2 

\^ + ifi X^ + ^ \^'f^+^l 
du dv ' dpi dqi ' dp2 dq2 

du dv ' dpi dqi ' dp2 dq2 : 

If X satisfies this equation the differential relations du = \dv, dpi = A^i, dp2 - Mq2 reduce to two only, 
since u, v, pi, ^i, p2, ^2 are connected by the equations 

/l = 0,/2 = 0,/5- 0. 

By integrating these two we find two more relations involving two arbitrary constants. Hence, wq ^niay 
suppose V, pi, p2y qi, 92 expressed in tenjils of t/, atid fin4 a solution by eliminating u from the following : — ' 

, u :^ piXi +p2X^ - y, 

V = ftJCl + 22»2 - ^, 

1 =. Xi dpifdu + ^n dp2ldu, 
VOL, CXCV. — ^A, 2 A 



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178 . MR. A. C. DIXON ON SIMULTANEOUS 

Then 
rf(«i» y) = pA^i> ^2) + [^x + ^J^i^uPi) + (ajj + ^J%i» P2) + g^ <H^u qi) + 

and similar expressions may be found for (/(ajgj y), c?(a;j, 2), ^(arg, 2) in terms of the 
bidifferentials of the pairs of independent variables. 

Let c„ ^2, C3, c^ be the constants of integration in a new complete primitive 
found by the method of §§ 27-8. Let X be the common value of the ratios 
dp^Jdqi^ dpjdq^, d^/d^. Then, after integrating the equations dpjdq^^ = 
dpjdq^ = d^/d^ (= X) by help of an assumed relation connecting, say, jp^, g^, jpg, g^, 
(?i, Cj we have four relations among 

and we may therefore suppose pi, pg* S'l* 5^2 expressed in terms of X, Cj, c^, C3, C4, 
unless X is a constant, and therefore itself a ftmdtion of Cj, Cj, C3, C4. Then 

dpi -^ X d^Tj, dp2 — X dq^^ d<ff '^ Xd^jt 

will be linear combinations of dcj, rfcg, cfcj, 0^4, and so will some such expression as 

adpi + $d\^ 

where a vanishes if X is one of the constants or a function of them. Conversely, 
rfcj, rfcg, dcg, dc4 will be linear combinations of 

dpy — X d^i, c?p2 " ^ ^9^2> ^ — ^ ^^> a ^JPi + /8 ^X, 

and the bidifferentials of Ci, e,, Cg, C4 in pairs will be linear combinations of the six 
following expressions : — 

^ {Pi> P2) -^ >^d{qv p%) - ^{Pi^ q^) + ^^^ (?i> qz)^ 

d (i>i, <^) - Xd (^1, <^) - Xd ( Pi, i/r) + fc«d (/?!, 1/1), 
^ (i>2> <^) - ^d (^2, <^) - \d {p^, i/r) + X^d (92, i/r), 
/8d (jpi, X) - akd {q,, p,) - fiXd (q,, X), 

a^ (i>2> Pi) + fid ( P2» ^) - a^^ (?2» Pi) - fi^d (^2, X), 
acZ(<^, p,) + i8d(<^, X) - aXdii^, p,) - 0XiJ(f, X). 

These are combinations of the bidifferentials of p^, p^^ q^^ q^y in pairs, with the 
expremonfl 

d{<^,X)-Xci{iA,X). 



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• PARTIAL WFPEBENTIAL EQUATIONS. 179 

Now X is a definite function of ojj, aug, pi, p^, qi, q^, given by eliminating the 
differentials from the equations 

dpi s= \dq\> dpi ^ A<ig2, d^ = )di^. 
By means of the first two, the third becomes 

and the fourth 

The result of ^^lioMiifiAion i« th^rofoise 

This shows the form of X as a function of asj, jCj, ^Ji, jjj, ^j, ^j, not involving Cj, Ca, 
Cj, C4. Now this choice of X makes it possible to choose coefficients A, B, C, E, F, G, 
such that 

^1 ^i + ^» dPi -\- d<f> = A {dpi — Xdg,) + B {dp.^ — Xdgj) + C {d^ — Xdi|»), 
Xi dqi+x^dq^ + d^li^E {dpi — kdqi) -f F (rfpa - Xtijj) + G (rf^ - Xdt^). 

Thus 

Hd{Pi, X) - \d{qi, X)} +B {rf(i>2. X) - Xd{q„ X)} 

+ C{d(<^, X) - Xdi^j., X)} = xid{p„ X) + a;ad(pj, X) + d{(f>, X) 

= multiples of bidiffereatials of jp^, p^, qi, q^ 

+ ^{«i <^ bi. ari) + Xid (pi, x^) + rf (^, a;,)} 

iW r * I 

+ a^{»i <^ 0>;, .aJ«) + XafiiPp x^ + d {^, aa)| 

~ a^l*^ ^^' ^») ~ P^^ ^^^' ^»q "*" ^l*^ ^^' ^^^ ~ ^^'^ ^^^' ^*q 

+ miiltiples of bidifferentials of ^], p^, q^, q^. 

2 A 2 



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180 MR. A. C. DIXON ON SIMULTANEOUS 

In like manner 

E {d(i)„ X) - Xd{q„ X)} + F {d{p,, X) - \d{q,, X)} 

+ G {d{<f>^\) - \d (i|r, X)} = ^ {d {z, x^) - g.,d (a;^, x,)} + ^-{d{z,x^) - q,d {x,, x^)} 
+ multiples of bidifferentials of p^, p^, g^, ^g. 

Hence the three expressions 

d{p,,\)^\d{q,,\), 

are all reduced to the same, save for a factor, by adding or subtracting multiples of 
the bidifferentials of Xj, a?^, ojj, x^ and of u^y %, 1^3, t^^ ; the same is therefore true of 
the bidifferentials of Cj, Cg, C3, c^. Hence all the new complete primitives found by 
the method of §§ 27-8 only add one to the eleven known " bifimctionally " indepen- 
dent pairs of functions satisfying the auxiliary equations ; one more pair, leading to 
a fresh complete primitive, is yet to be found. 

§ 30. These results may be used to construct examples of bifunctions. For 
instance, the equations 

% = q^x^ + q^^ + p^y 

lead to the following case among others : — 
In the equations 

^ = ^ = -^^^ = X 
dq^ dq^ dp^ 

put (/o = X + a, dq^ = rfX, and integrate. 

Thus 2^2 = X* + 6, 351 = X' + c, 4pi = X* + e, 

and the arbitrary constants a, 6, c, e in the new solution are respectively equal to 

q,2 — X, 2^2 — X^, Sq^ — X^, 4pi — X*, where X^ajj + Xg + X = 0. 

Now from § 29 it follows that d (c, e) can be expressed in terms of d (a, 6), the 
bidifferentials of x^, x^, y, z and those of p^, p^y ^u 9^- 
For convenience, let us write 

w, V, w, Xy y, z for Xj, X, py p^y qi, q^ respectively ; then 
for x^ we must put — v(l + wv), 
for y „ „ tw — xv{l + uv) + y, 

for z „ „ yu — zv(\ + wv) + x, 



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PARTIAL DIFFERENTIAL EQUATIONS. 181 

so that the eight original variables connected by two equations are now expressed in 
terms of six. 

Thus d (3y — t;^, 4ti; — v*) can be expressed in terms of d (2; — y, 2a? — v^), the six 
bifferentials of Wy a:, y, z and those of 

u, — t;(l + uv\ wu -— xv{l + uv) + y, yu ^ zv{l + uv) + x, 

that is, of 

u, V, wu — xv{l + t^v) + y, yte — zv{\ + wv) + x. 

There is no difficulty in finding the relation. It is 

M»(i(3y - t;», 4i^ - t;4) - 62;2(1 -f uvfd{z - i;, 2x - v^) 

- 12w2rf(y, w) + 12v2(l + wi;)»d(z, x) 

+ 12i;*{(l + ^v) {y — 2;t;*) ^ ^(^ — ^)}d{Vy u) 

— I2i;*{l + uv)d{Vy yU'^zv '^ uz't? + x) 
+ 12i^t;*c?(v, t(n^ — ajv — t^on;^ -f y) = 0. 

Here then we have an identical linear relation connecting the bidifferentials of 
seven pairs of fiinctions of six variables. Any one of the seven pairs is accordingly 
by definition a bifimction of the other six. 



Second Application. 
§ 31. Take now a differential equation of the second order, 

where p, q are the first and r, 5, t the second partial derivatives of z with respect 
to Xy y. 

A complete primitive will consist of a single equation in aj, y, z involving five 
arbitrary constants, say aj, ag, ^3, a^ a^. If we form the first and second derivatives 
of this equation we shall have, in all, six equations from which a^, aj, aj, a^ a^ can 
be foimd in terms of a:, y, 2;, jp, q^ ?% 5, ty and the original differential equation will be 
the result of eliminating a^, a^, ag, a^y a^. Let w^, w^, Ug, -w^, Wg represent the 
expressions found for a^y a^y %, a^, % respectively, in terms of Xy y, «, p, y, r, 5, t. 

Then from the equations 

/=0, Wi=ai, Wgsa^, 

by differentiating, we can form six equations which will involve the third derivatives 
of 2 ; by eliminating these we deduce the following two differential equations to be 
satisfied by Wj, u^ : — 



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182 MR. A. C. DIXON ON SIMULTANEOUS 

J{x, r, t) + pJ{z, r, ^ + rJ(jp, r, «) + s3{q, r, ^ + J(y, «,-<) + g^^ «, e) 

J(a?, «, r) + pJ(2, 5, r) + rJ{p, «, r) + 8J{q, Sy r) + %, i, r) + 5J(2:, ^, r) + sj{p, i{, r) 
+tJ{q,t,r) = 0. 

Here J ( ) denotes the Jacobian of /, u^, Uc^ with respect to the variables specified. 
These equations express the conditions which are necessary and sufficient in order 
that 

dz = pdx -f qdyy 

dp = rdx + sdy, 

dq 9 3ch + idy 

may be integrable without restriction, when r, 5, ^ are given in terms of Xy y, 2;, |?, g', 
by the equations 

the conditions must of course be satisfied by any three of the six functions Wj, Wg, 
tig, %, tig, yi We thus have forty equations, of which only eight can be algebraically 
independent 

I '32. The conditions to be satisfied by ti^, u^ are linear and homogeneous in their 
Jacobians with respect to the eight variables x, y, 2J,p, g, r^ s, t; of these, one is 
given in terms of the rest by the equation /= 0, and may, if convenient, be sup- 
posed not to occur in u^^ u^ : hence the auxiliary equations in this case have seven 
independent variables and the dependent variables do not occur explicitly: to 
find a solution we are thme&r^ to Soim a OQmpkte bidi£S^WPtial, whioh ^all bp a 
linear combination of the determinants of the following array : — 



d{r,s), 





-X-^Z-rP- 


-sQ 


d{r,ff, 


Ti + pZ + rf + iQ, 


_ Y - ^ - ^P - 


-tQ 


d{s,t), 


Y + qZ + sP-\- tQ, 







• (^P.^, 


rT, 


-rS-AT 




(5) d{p,s). 


*T, 


rJR 




dip.,tl 


- rB - «S 


sB. 




«?(2. r). 


sT, 


-S-tT 




d{q,s), 


tT, 


sR 




d{q, t), 


- sR - <S, 


tR 




(10) d{z,r), 


i>T, 


-pS-qT 




d{z,s). 


?T, 


pB. 




#,<9. 


-/)R-^, 


^R 




d(x, r), 


T. 


-S 




d{x, s). 


0, 


R 





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PARTIAL DIFFERENTIAL EQUATIONS. 183 



(15) 4x,<), 


-R 









^.r). 


0, 






-T 


%.*)» 


T, 









%.o. 


-s. 






R 


Ax,p), 


0. 









(20) d{9,pl 


0, 









<H?^>pl 


0. 






0- 


rf(x,Qr), 


0, 









%.^). 


0, 









d{z,q), 


0, 









(25) d(p,g), 


0. 









ci(x,«), 


0. 









%.^). 


0, 









d{x,y\ 


0, 









X,P. . 


. are written for 


df/dx, 


,^fl^ . . . 





Of these twenty-eight rows, only twenty-one are independent* Fw instanoe, 
multiply the Ist, 2nd, 4th, 7th, 10th, 13th, 16th by - S, - T, P, Q, Z, X, Y respec- 
tively and add ; the resulting row is 

d(/r), 0, 0, 

which vanishes since f=Ohy hypothesis. 

Suppose d {uyy u^) to be the complete bidifferential formed from the determinants 
of the array, then to complete the solution we have to find r, 5, t from the 
equations 

and integrate the equations 

dz = pdx + qdy, dp = rdx + sdy, dq = sdx + tdy. 

It will amount to the same thing if we treat t*i as known in the auxiliary 
equations. They must be satisfied if Wg, u^, u^ are substituted in turn for tig. Now 
two homogeneous linear partial differential equations in seven independent variables 
can at most have five common solutions, and here one of these, tt^, is known ; the 
other four may be taken as u^, t^, ^4, u^. 

§ 33. Any two of the five ftmctions x, y, 2, p, q will satisfy the auxiliary equations, 
but as we have to solve for r, 5, ^, these solutions will not serve our purpose. They 
are ten in number, and ten more will be given by taking in pairs the expressions ti^, 
^2> ^'3> ^4> '^5 given by any complete primitive. These twenty are not all bifimc- 
tiotially independent, for since there are three relations* among the ten expressions 

a?, y, 2J, p, q, Wi, 1/2, W3, «4, z^6, 

* Compare § 34> {k 184. 



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184 MR A. C. DIXON ON SIMULTANEOUS 

three linear relations can be formed connecting the twenty bidifferentials ; one is 
formed from each pair of equations as at § 17 (8). Hence seventeen biftmctionally 
independent solutions of the auxiliary equations are known when we have one com- 
plete primitive. The ftdl number is nineteen (t^ " 2 j, and in order to know all 

we must have one, or possibly two (see § 41, p. 190), more complete primitives. 

§ 34. New solutions found by varying the parameters may be divided into two 
classes, according as the parameters are or are not all fiinctions of one variable ; 
solutions of the former class only occur in exceptional cases, and the principles of § 21 
apply to them with slight modification. 

Let the three equations connecting 

«i Vf 2J, p, q, tti, Wg, ^8. ^4* ^6 

be <i>t{x, y, z, p, q, u^, u^, Wj, W4, u^) = (t = 1, 2, 3) ; 

(the forms <^|, <^2, ^g are not unrestricted, but must be such that the following rela- 
tions hold identically 






or we may take <^i as not involving />, q and ^^^^ P d<f>Jdz + 9<^,/3ic 
then the variations of the parameters must satisfy the three equations 

[l^^d^r ^ {i = 1, 2, S), 

in order that the same relations may subsist among a?, y, 2, p, q, r, s, t and the para- 
meters, as held when the parameters were constant. 

If the parameters are functions of one variable, their forms must be so chosen that 
the three equations last written reduce to one only, otherwise we shall have five 
relations connecting x, y, z, p, q with this single variable. 

§ 35. If the parameters are not functions of one variable, only the equations 

are equivalent to six, and determine the partial derivatives of v^, u^ u^ with respect 
to Wj, % in terms of the five parameters and aj, y, 2;, jp,, q. By help of the relations 
^i = we may suppose x, y, 2, p, q eliminated and thus arrive at a system of four 
partial differential equations connecting Wj, u^y w,, w^, u^. 



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PARTIAL DIFFERENTIAL EQUATIONS. 185 

The original system may also be taken to consist of four equations connecting five 
variables a:, y, z, p, 9, namely : 

dzjdx = p, dzldy = 5, dp/dy = dq/dx 
f{^y y. 2J, p, g, dp/dx, dp/dy, dq/dy) = 0, 

and so the method of variation of parameters does not lead to any simplification of 
the problem in general. 

§ 36. The interchange of variables and parameters is again possible ; it is, perhaps, 
made clearer by taking three equations of perfectly general form, 

<f>i (o^i, x^y x^y x^, x^y u^y u^ u^, u^, Ug) = (^ = 1 , 2 , 3), 

connecting two sets, each of five quantities. 

Whichever set we suppose constant and eliminated by diflferentiation, we are led 
to a system of four partial diflTerential equations connecting the quantities of the 
other set, two of the five being taken as independent variables. A new solution of 
either of these systems of differential equations will in general yield a new solution of 
the other. 

Suppose, for instance, that we have a new solution of the u equations ; this gives 
t^3, u^ Wg, say, in terms of f.^, u^. Then the six equations included in 



't^^' dur- 0(i= 1, 2, 3) 



give two relations among cc^, . . . Wj, t^^, since the four differential equations, 
which are consequences of these six, are supposed satisfied ; by the help of these 
two, u^y u^y may be eliminated from the three relations <^i = 0, ^^ = 0> ^3 == ^> ^^^ 
thus three relations are given connecting aj^, iCg, x^y x^y x^ ; these three will constitute 
a solution of the x system of differential equations. 

§ 37. In this more general case there will not seemingly, as a rule, be any more 
solutions for either system of differential equations. For the derivatives, say, of 
^> ^4> ^5 with respect to x^y x^ are given in terihs of these five variables and two 
others, say Mj, u^. The forms we may assign to u^y u^ are then restricted by three 
differential equations derived fi'om the three conditions 

-i^ J^'^L^ /'r — 3 4 ^) 

dx^ cLc^ ■" cU^ clx, ^ "" ""' ^' ""'' 

and thus, generally speaking, no forms of W|, iig will be suitable. In some cases the 
conditions are not inconsistent, and we may form an array by the method of § 11 
such that if d{0yx) is a combination of its determinants, then ^ = a, x = &> ^1 = 0. 

VOL. CXCV.— A. 2 B 



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186 



MR. A. C. DIXON ON SIMULTANEOUS 



<^2 = 0, ^3 = Mrill give suitable values for Wi, Wj. This array will have four columns 
and forty-five rows, ten such as 

d{xi,xj), 0,0,0, 
ten such as d (m„ uJ), 0, 0, 0, 

and twenty-five of the following type. ■ In the first column there is d{xi, Uj), in the 
(r + l)th the minor of S^<l»r/dx,duj, in the determinant 



d^<f>r a^r 3'^r 



^<l>r 



3*^r 



d.i\ dtii 3^3 dii^' Sij 3mj' B'j 9«j' 8/j 3j<j 



3'4»r 

3dii 
3r^ 

3^s 

3^1 



3^ 

3*3 

3^8 



3^ 

3«8 

3^ 

3^3 
3*3 
3tB, 



for 



3^ 
3^, 

3^3 

3c^ 
r = 1, 2, 3, 



3^<^, 

a^j 3»<j 

3^ 
3<ib 

308 
3^5 



3<j^ 
3^ 

3^8 

3*1 
3z^3 

3ii 

3^4 

3^ 

3l^6 



3*2 3*3 

9mj 3?^i 

3*3 3^ 

3*2 3*3 

3ws 3i^3 

3*2 3(^g 

3*2 3^ 

3^6 3i^6 






















This interchange of variables and parameters may take place whenever their 
numbers are equal, the diflferential equations being of the first degree. 

Eooamjyles. 

§ 38. L As an example of the method of solution take the equation a = /8, where 
a is a function of r, 5, p — sy^ x and fi a function of s, t^ q ^ sx, y. 

In the array (§ 32) multiply the first row by 9a/9r, the fifth by 9a/9jt>, the fourteenth 
by 9a/9a:, the seventeenth by — 5 9a/9p, and add ; the resulting row is 

d{a, s), 0, 0. 
Hence we take a = /8 = a, 5 = 6, 

z = hxy + X + Y, 

X being a function of x only and Y a function of y only. Then a = a is a relation 
connecting x, dX/dx, d^X/dx^, and fi = a is a relation connecting y, clY/dy, d^Y/dy^^ 
and by solving these for X, Y respectively we shall have the complete primitive. 



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PARTIAL DIFFERENTIAL EQUATIONS. 187 

§ 39. IL As a second example take the equation 

F(r, s,t,p — ^,q — ty,z — qy + \tif^, or) = 0. 

Here the third row of the array is 

d{s, t\ 0, 0, 

so that the fiinctions «, t satisfy the auxiliary equations. Put, then, 5 = a, t = 6 ; 

thus 

gr = owr + 6y + c 

2 = aa?y + i 5y» + cy + X, 

the last term heing a function of x only. The differential equation thus becomes 
F{d^X/clx\ a, 6, dX/dx, ax + c, X, x) = 0, 

an ordinary equation of the second order giving X in terms of x and two more 
arbitrary constants ; hence the finding of a complete primitive is reduced to the solu- 
tion of the equation last written, 

§ 40. ni. If the equation is of the particular form F(r, s, t,p ^rx ^sy,q — sx^ ty, 
2: — px — qy+^rx^+sxy+^ty^) = 0, the first three rows of the array are 

d{r,s) 
d{r,t) 
d{s, t) 0. 

Hence any two of the three functions r, s, t will satisfy the auxiliary equations, 
and a complete primitive is given by putting 

r ^= a^ s = h^ t = b. 

Hence p = ax+hy'\-g, q = ^+6y+/ 

z = c + gx +Jy + ^{aa^ + 2hxy -f bf), 

where a, 6, c,/, g^ h are constants satisfying the relation 

This is a case in which other solutions are readily given by supposing the para- 
meters variable and functions of one variable only, say a. The variations must 
satisfy the conditions 

x^da + 2xydh + yHh + 2xdg + 2ydf+ 2dc = 0, 

a: c?a + y rfA + dgr = 0, « dA + y d6 + cZ/ = 0, 
2 B 2 



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188 ME. A. C. DIXON ON STMITLTANEOUS 

whence follows xdg'\-ydf-\-2dc — 0, a simpler relation that may be taken instead of 
the first of the three. 

These equations will define x, y in terms of the single variable a, unless all the 
first minors of 

vanish. 



dkt 


dh 


dg 


dh 


dh 


df 


dg 


df 


'idc 



We thus have three ordinary differential equations connecting a, 6, c, f^ gyh\ 
they are connected also by the relation F (a, ^, 6, g, /, c) = 0, and the fifth relation 
among them may be chosen arbitrarily, so that we may put h = (j) (a), an arbitrary 
function. 

Then we have 

db/da = {f (a)}^ df/da = f (a) dg/da, 
2dc/da = {dg/daf, 
F{a,<f>{a),b,g,f,c) = 

as the equations determining 6, g^ /, c in terms of a. These are to be integrated, 
and then a is to be eliminated from the equations 

03 + y dhjda + dg/da = 0, 
z = c + gx +fy + ^{ax^ + 2hxy + hy^). 

The result of elimination will be a solution of the differential equation. Three 
constants are introduced by integration, and thus, if the function ^ involves two 
constants, the new solution will generally be a complete primitive. 

§ 41. The new complete primitive gives new solutions of the auxiliary equations 
which we shaU now examine. Let a^, ag, Og, a^, a^ be the new set of parameters. 
Then a, h, g, 6, c, / are connected with these parameters by five equations, one of 
which is the original equation F = 0. These five relations are such, that if 

dh = Xrfa, dg = /jida , 
then db = \^da^ 2dc = /ji^cZa, df = Xfxda ; 

of these five, the first two define X, fx in terms of a, a^, a^, ag, a^, ag, and the others 
must then follow from the five equations that give A, g, 6, c, / in terms of a and 
the same new constants. Thus, in general, we may suppose a, A, g^ 6, c, f, /ji, 
expressed in terms of X, a^, a^, ag, a^, a- and the expressions will be such that 

dh — \da^ dg — [xda, db — \^da, 2dc — fjL^da, df — Xfida 



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PARTIAL DIFFEREXTIAL KQrATIOSa 189 

involve only the differentials of «!, a^ fc^, •♦* *5- ^^ ^f these five is expites^hle in 
terms of the other four, since 

cF cF ?F rF rF rF 

da * ck ' c* o/ ^ </^ • flr 

while one of the relations connecting X, ft, a, 6, ... is 

aF , ^ar , pf , ^.?f , ^ aF , , ^?f 

Some expression such as vd\ — pda will also involve the differentials of a^, 04, a^, 
a^, ttj only. Hence the differentials of aj, a^, o^, a^ a^ will be linear combinations ot 
vdX — prfa, dh — Xc/a, rf^ — firfa, c/6 — X-t/a, rf/*— X/jw/a, 2dc — fiVa, of which the 
last five satisfy a linear relation. 

Thus the bidifferentials of a^, a^, a,, a^, a^ in pairs will be linear combinations of 
the bidifferentials of a, 6, c, /, gr, A (only five of the six need be used) in pairs^ and 
of the expressions 

d{h, X) - Xrf(a, X), %, X) - ^d(a, X), c/(5, X) - X«rf(a, X), 
d{f. X) - X^rf(a, X), 2d(o, X) - /^^^(a, X), 

of which last five, only four are independent. 

Now X, II are connected not only by the equation 

dF dF dF dF dF QF 

^ + ^a^+ '*^+ ^V+ ^37 + ^/^V = ^' 

but also by the equation 

35 + Xy 4- /t = 0, 

so that they are definite functions of x, y, a, h, /, g, h. 

Again p = ax + hy + g, 

d{p, x) — hd{y, x) = xd{a, x) + yd{h, x) + d{g, «), 

«^(p. y) — «<^(». y) = ^<^(«. y) + W(^. y) + %. y)- 

Thus y{<f(A, X) - Xrf(a, X)} + {%, X) - ftd{a, X)} 

= xd{a, X) + yd(A, X) + %, X) 

= a^ [^(p. ^) - ^«'(y. a')] + aj; I^^p^ v) - ^K^> y)l 

+ multiples of bidifierentials of «, 6, c, /, g', h. 



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190 MR. A. C. DIXON ON SIMULTANEOUS 

In like manner 

y{d{h, X) - X^a, X)} + {d{f, X) - \iid{a, X)} + x{d(h, X) - \d{a, X)j 
= xd{h, X) + yd{h X) + d(y, X) 

= a^ \d{<i^ ^) - &%. «^)] + ^ \A{^^ y) - ^d{x, y)\ 

+ multiples of bidifferentials of a, 6, c, yj gr, h. 
Lastly, 

2xy{d(h, X) - Xrf(a, X)} + y*{(^(6, X) - \H{a, X)} + 2ir{%, X) - ixd{a, X)} 
+ 2y{d(/, X) - X,irf(a, X)} + {2ti(c, X) - /.V(a, X)} 
= a?d{a, X) + 2xyd{h, X) + y^d(6, X) + 2xd{g, X) + 2t/d(/, X) + 2rf(c, X) 
= 2{d{z, x) — {hx + by +f)d{y, x)}d\/dx 
+ 2{d{z, y) - {ax + % + 5r)d(^, y)} d\/dy, 

+ multiples of bidifferentials of a, 6, c, /, g^ h. 

Hence, in all, nine combinations of the ten bidifferentials of a^, ag, ag, a^, a- can 
be expressed in terms of the bidifferentials of x, y, z, p^ q and of a, 6, c, fy g,h\ that 
is, in terms of the bidifferentials of the seventeen known independent pairs of functions 
satisfying the auxiliary equations : thus the new complete primitive adds only one to 
the number of these known bifunctionally independent pairs, and one more must be 
added in order to give the full number. 

This theory enables us again to construct examples of bifunctions of a number of 
known pairs which may reach eighteen. 

§ 42. The foregoing investigation may be modified so as to give singular solutions 
of a pair of differential equations of the form in question, say 

Fi(r, s, t, p, q, z) = 0, 
F^Cn s, t, p, q, z) = 0, 

where p = p -- rx ^ sy, 

q = q — sx --ty, 

A complete primitive would be given by supposing r, 5, t, p, q, z constants con- 
nected by the above equations. Another solution would be given by solving the 
total differential equations found by supposing the relations 



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PARTIAL DIFFERENTIAL EQUATIONS. 191 

xdr -{-y ds + dp = 0, 
xds '\' y dt -^ dq = 0, 
xdp+ y dq + 2dz = 0, 

to reduce to the same relation linear in x and y. That is, we must solve the system 

ds = \dr, dt = X^dvy dp = fidr, 
where X, /x are given in terms of ^, q^ 2, r, s, t by the relations 
9Fi 9Fi vaSFj 9Fi 9Fi ^ 1 o9Fi 

ar + V + ^V + ^r + ^'^ar + ^'^ar =^' 

9Fg 9F3 xo9F 9F3 9F3 - 2 9F, 

and 5^, 2J in terms of jp, r, 5, i by the relations F^ = 0, F2 =0. 

The complete primitive of these ordinary equations will involve three arbitrary 
constants, and there may be singular solutions with a lower number ; none of these 
will therefore constitute a complete primitive of the partial diiSferential system 

Fi = 0, F3 = 0. 



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[ 193 ] 



V. The Velocity of the loris ^produced in Gases by Rmtgen Rays. 

By John Zeleny, B.Sc., B.A., Assistant Professor of Physics, University of 

Minnesota. 

Communicated by Professor J. J. Thomson, F.R.S. 

Received February 5, — Read March 1, 1900. 

§ 1. Introduction. 

The electrical conductivity which is imparted to gases by their exposure to Rontgen 
rays has been explained by J. J. Thomson and E. Rutherford* on the hypothesis of 
a formation of oppositely charged carriers throughout the volume of the gas. The 
motion of these carriers or ions when in an electric field constitutes the observed 
conductivity, and the recovery of the insulating property of a gas after an exposure 
to the rays is due partly to the recombination of the oppositely charged ions and 
partly to their impact with the boundaries. 

An estimate of the sum of the velocities with which the positive and negative 
ions move in air when in a unit electric field was first obtained by J. J. Thomson and 
E. Rutherford, and later E. Rutherford,! by the same indirect method, determined 
the sum of the velocities of the ions in a number of gases. This method involved 
the determination of the rate of recombination of the ions, the saturation current 
obtained through the gas by the use of a strong electric field, and the current 
obtained with some small non-saturating electric force. E. Rutherford also 
describes an experiment in which the velocities of the two ions in air were obtained 
separately by a direct method, and found to be approximately equal. The writerj 
has since shown that in general the two velocities are not equal, and for those gases 
for which the ratio of the two velocities was determined the negative ion moved the 
faster in nearly all cases. 

The values of the velocities of the ions have recently been applied by J. J. 
Thomson§ and J. S. Townsend|| in the determination of important physical quantities, 
and it seemed desirable that a redetermination of the values of the velocities be 

* J. J. Thomson, and E. Rutherford, * Phil. Mag.,' November, 1896 
t E. Rutherford, * Phil. Mag.,' November, 1897 
X J. Zeleny, * Phil. Mag.,' July, 1898. 
§ J. J. Thomson, *Phil. Mag.,' December, 1898. 
II J. S. Townsend, *Phil. Trans.,' A, vol. 193, 1899. 
VOL. cxcv— A 266. 2c 9.11.1900. 



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194 



MR. J. ZELENY ON THE VELOCITY OP THE IONS 



undertaken, partly because of advances in our understanding of some of the intrica- 
cies of the conduction, and partly because it seemed desirable that a satisfactory 
direct method be devised whereby the velocities of the two ions could be determined 
separately, and in which the experimental conditions could be subjected to a number 
of variations sufficient to ensure freedom from serious errors. 

In undertaking this, an attempt was first made to use a modification of the method 
employed by the writer in the determination of the ratio of the two ionic velocities, 
which is described in a previous paper. The ions were made to go against a stream 
of gas in a tube by means of an electric field, and their velocity was compared to that 
of the gas stream. The presence of the gauzes necessary for the production of the 
electric field was found, however, to disturb the gas stream sufficiently to produce a 
turbulent motion in it and so prevented the attainment of absolute results. 

The method which was then developed, and the one with which all of the results of 
this paper were obtained, also consisted in directly comparing the ionic velocity with 
that of a stream of gas, but avoided the difficulty of the above by having the electric 
field at right angles to the gas stream. 

§ 2. The Method Used for Determining the Velocity. 

A stream of gas is passed between two concentric cylinders which are kept at 
difierent potentials, and which at one place are traversed by a beam of Rontgen rays. 
The ions which are produced between the two cylinders by the rays are carried 
along by the stream of gas and at the same time, under the influence of the electric 
force, they move at right angles to the axis of the tubes. The resultant paths of the 
ions are inclined by an amount depending upon the relative value of the velocity of 
the gas stream to that of the ions. 

Let CC in fig. 1 represent a section of a portion of the outer cylinder, and DB 
that of the inner one, and let dd represent a narrow beam of rays traversing the two 
cylinders at right angles to their common axis. When the two cylinders are at 




Fig. 1. 



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PRODUCED IN GASES BY RONTGEN RAYS. 195 

different potentials and the gas between them is at rest, an ion starting from the 
point d at the inner surface of the outer cylinder will move directly across to / 
imder the electric force. But when a stream of gas is passing between the cylinders 
from right to left, the ion will also be carried along by the stream, and so follow a 
path somewhat like that represented by the curve dk^ finally reaching the inner 
cylinder at some point, i, which can be determined. The paths of the ions are not 
straight lines, because the electric intensity and the velocity of the gas stream vary 
from point to point between the cylinders and according to a different law for each. 
The distance X that the ions have been carried along the tube by the gas stream 
while they are crossing between the two cylinders under the electric force is a 
measure of the relative velocities of the gas and of the ions, and so may be used in 
determining the velocity with which the ions move in a given. electric field. 

Let the outer cylinder be kept at a potential of A volts and the inner one at zero 
potential. 

Let h be the inner radius of the outer cylinder and a the outer radius of the inner 
cylinder. 

Then the potential at any point between the cylinders at a distance r from the 
common axis of the two cylinders is 

and the electric intensity at this point is 

dR A 



dr rlogebfa ' V /• 

If we let V represent the velocity with which an ion moves when in an electric field 
whose intensity is 1 volt per centim., and assume that its velocity is proportional 
to the strength of the field, then at a point whose electric intensity is represented bv 
equation (2), the radial velocity of the ion will be 

V = -^ . (3). 

The ion being carried by the moving gas also has a motion along the tubes. The 
velocity of the gas stream at any point depends upon its distance from the axis of 
the cylinders, which will be called the x axis. 

Suppose that at the distance r from this axis the gas velocity is u. 

The motion of the ion is represented by 

dx u 

~dT~ Y ' ' ' \ (^/' 

and substituting the value of V from (3), 

2 c 2 



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196 MR. J. ZELENY ON THE VELOCITY OF THE IONS 

The distance X travelled by the ion in the direction of the x axis while it is 
traversing the whole distance between the cylinders, i.e., from r = 6 to r = a, is 

^='^:'"f/-^'- («)• 

Now the average velocity of the gas stream as measured by the quotient of the 
total volume of gas emitted in a second by the area of the cross section is 

From (6) and (7) 

^= 2kv ^^»a • • • (^)' ^""^ ^= 2AX ^^g-a " * ' ^^^^ 

This gives the value of the ionic velocity in a unit field in terms of quantities which 
can be experimentally determined. 

The time required for the ions to pass from one cylinder to the other is 

The equations above apply to ions starting from the inner surface of the outer 
cylinder and moving inward to the inner cylinder. In practice it is not possible to 
limit the production of the ions by the Rontgen rays to the inner surface of the outer 
cylinder, so a narrow beam of rays is passed at right angles through the cylinders, as 
is represented by dd of fig. 1. Of the ions of this layer which move inward under 
the influence of the electric force, those that start from the circumference at d are 
carried the farthest by the gas stream before they reach the inner cylinder. Under 
these conditions the equations obtained can be applied by determining the point along 
the inner cylinder farthest from the beam of rays that is still reached by ions. For 
obtaining this point, the inner cylinder DB is divided at k into two parts, insulated 
from each other, the part B to the right being connected to earth, while the part D, 
to the left of the division at i, is connected to a pair of the quadrants of an 
electrometer. 

If a definite stream of gas is maintained between the two cylinders, then while the 
potential of the outer tube CC is above a certain value, all of the ions from the 
volume dd which move inward will reach DB to the right of the juncture k, and so 
the electrometer reading will not change. By gradually diminishing the potential of 
CC a value is finally reached such that the ions starting from the outer edge d reach 
DB just to the left of k, as wiU be indicated by a changing electrometer reading. The 
value of the voltage A in equation (9) is thus determined, and the value of X, which 
corresponds to it, is the distance from the beam of rays to the juncture k. In getting 
X the corrections which must be made for the width of the beam of the rays and for 



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PEODUCED IN GASES BY RONTGEN RAYS. 



197 



the width of the juncture k will be considered later. The apparatus as used will now 
be described. 

§ 3. The Apparatus. 

The main parts of the apparatus are represented in fig. 2, where the lower part of 
the figure is a vertical section, while the electrical connections in the upper part are 
viewed firom above. 




Fig. 2. 

The outer cylinder, A A', had an internal diameter of 5*11 centims., and a total 
length of 142 centims. For convenience the length is shortened in the figure by the 
omission of two sections. The part to the left of V, 41 centims. long, and the part 
to the right of V, 81 centims. long, were made of strong brass tubing. The portion 
DD' between these was 20 centims. long, and consisted of an aluminium tube, which 
was of the same internal diameter as the brass cylinders. Brass collars over the ends 
of the aluminium tube fitted into the external collars V and V soldered to the brass 
cylinders, and so formed close-fitting joints that were made gas-tight by sealing them 
on the outside. The whole cylinder was supported on a board, XX', and insulated by 
means of four paraffin blocks, two of which are represented by P and P'. 

The inner cylinder, BB', was an aluminium tube 1 centim. in diameter, closed at its 
ends by conical pieces. At C the cylinder was divided so that the two portions were 
held one-half of a millimetre apart and insulated, by means of an ebonite plug. At 
the end, B', the tube was supported and kept central by means of two small ebonite 
rods, Q. The tube was further supported by the two stiff brass wires, Y and Y', 
which lead through the ebonite plugs, R and R', in the outer cylinder, and served to 



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198 MR. J. ZELENY ON THE VELOCITY OF THE IONS 

make electrical connections. The part B' was joined to earth, while the part B w^as 
connected to a pair of the quadrants of the electrometer, E. Great care was taken to 
adjust the position of the central cylinder so as to be accurately concentric with the 
outer one. 

The ends of the outer cylinder were fitted with the large rubber stoppers F and F'. 
Through these passed the gas inlet and outlet tubes, whose ends were the elongated 
funnels J and J'. These funnels, together with the cone endings of the inner 
cylinder, made the lines of gas motion change less abruptly on entering and leaving 
the apparatus, and so aided in having the gas maintain a steady motion in DD', where 
the observations were taken. At the left end, F, a rubber tube led to a gas bag of 
about 150 litres capacity. The manometer, I, measured the pressure of the gas in the 
apparatus. The right end, F', was connected to the glass wool chamber, G, which 
served to remove dust and any stray electrification from the gas. A rubber tube then 
led to a drying or moistening apparatus, to be described later, which was connected to 
a large gasometer of the ordinary type. The pressure of the gas in the gasometer 
was measured by means of a manometer, and a scale was also attached to the 
gasometer for measuring its rate of descent during an experiment. The average 
velocity of the gas stream in the apparatus was determined from the volume emitted 
by the gasometer in a second, and from the area of the cross section between the two 
cylinders. To prevent the gas in the gasometer from getting moist too rapidly in 
those cases where dry gases were used, the surface of the water was covered with a 
layer of oil, such as is used for air pumps, because of its very low vapour pressure. 

The board, XX', with the attached cylinders was placed on the top of a lead- 
covered box, UU', so that DD', the aluminium portion of the outer tube, was above 
the aluminium window, W, in the box. 

The box contained the Crookes' tube and the induction coil for operating it. The 
form of tube used was that which the writer has previously employed for similar 
work.* This form was more satisfactory than any of the others tried, and gave the 
best results when emitting weak rays, and when an interval of rest of at least three 
or four minutes was allowed between the periods of use, which did not exceed thirty 
seconds. A 6-inch Apps' coil was used with a hammer interrupter, which could be 
made to run with sufficient uniformity with an easy running weak ray tube. The 
source of the rays, T, was more than 20 centims. from the axis of the cylinders. 

The narrow vertical beam of rays wliich was sent up through the cylinders was 
regulated by adjusting the position of the tube, T, and of the lead plate, S, with its 
narrow slit, and of the two lead rings, L and U, which fitted over the cylinder, DD'. 
This adjustment was first made by geometrical arrangement, and then tested and 
completed with the aid of a fluorescent screen placed over the apparatus. The lead 
strips, H and H', served to restrict tlie window, W, and the lead cover, Z, prevented 
any rays or ionized gas from reaching the outside air of the room. 

♦ J. Zeleny, a^hil. Mag./ July, 1898, p. 126. 



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PRODUCED IN GASES BY RONTGEN RAYS. I99 

The quadi'ant electrometer, E, used for making the measurements was a small 
bicellular one, the needle of which was suspended by a quartz fibre, and charged 
through the liquid below by means of a battery of 160 small storage cells. One 
pair of its quadrants was joined by a wire to the part BC of the inner cylinder. 
Both the electrometer and the connecting wire were surrounded by an earthed 
metal case. 

The key, K, permitted the insulated quadrants to be connected to earth at any 
time. 

The capacity of the two quadrants and the part of the inner cylinder connected to 
them, together with the connecting wire, was about 53 centims. The sensibility of 
the electrometer was about 500 divisions per volt, with the scale at a distance of 
130 centims. The potential of the outer cylinder A A' was maintained at any desired 
value by means of the battery of storage cells, N ; the arrangement of the extra 
cell, O, and the divided megohm, M, permitting the addition of a fractional part of a 
cell's voltage. 

By opening a stop-cock on the gasometer the gas was made to pass from the 
gasometer, through the apparatus, into the gas bag on the other side, at a rate which 
was regulated by the weights on the gasometer. It could then be forced back into 
the gasometer and used again. 

A large volume of gas is required for carrying out an experiment, and the method 
is therefore limited to a small number of gases that can be obtained in such quantities, 
and that do not act upon the materials of the apparatus. 

§ 4. CORRKOTIONS AND PRECAUTIONS OBSERVED IN THE EXPERIMENTS. 

1. It is essential for these experiments that in its motion down that part of the 
tube where the observations are being taken, tlie different portions of the gas should 
move in paths parallel to the axis of the tube, i.e., that the motion be uniform, and 
not turbulent with vortices. This condition depends upon the velocity of the gas 
stream. 

O. Reynolds has shown* that for motion in a cylindrical tube a fluid when 
started in a turbulent state wilj tend to assume a uniform motion with the parts 
moving parallel to the axis when for the fluid the average velocity is less than a 
critical value. 



V = 



A^ 



BpD' 

where /x is the viscosity of the fluid relative to that of water at 0°, p is its density, 
D is the diameter of the cylinder, and B is a constant. 

The value of B obtained was about 280 when D and V were measured in metres. 

Applying this constant to the gases used, for a cylinder of the diameter of the 

♦ O. Reynolds, * Phil. Trans.,' A, 1883. 



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200 ME. J. ZELENY ON THE VELOCITY OF THE IONS 

outer one in the apparatus, we obtain for the value of the critical velocity for air. 
about 55 centims. per second, and for hydrogen about 390 centims. per second. It is 
evident that in the apparatus used where there are two concentric cylinders, the 
maximum velocity consistent with a uniform motion must be considerably larger 
than if the gas were flowing through the outer cylinder alone. Nevertheless the 
largest value of the velocity used in any experiment was 25 centims. per second for 
air and 44 centims. per second for hydrogen. As these values are well within the 
limits given above for a cylindrical tube whose radius is equal to that of the outer 
one here used, the conditions for a stable motion are fulfilled. The entrance of 
the gas through a funnel-shaped aperture and its subsequent passage for a con- 
siderable distance through a uniform section allowed the motion to come to a per- 
manent state before it reached the place where the observations were taken. 

An experiment which was tried showed that by blowing a stream of air down a 
large glass tube and with a velocity greater than that used in these experiments, 
the gas assumed a motion parallel to the axis after it had traversed but a short 
length of the tube, as was made visible by the presence in the air of irregularly 
distributed ammonium chloride particles. 

2. The volume of the gas emitted per second by the gasometer varied a little for 
different elevations of the gasometer, but there was a considerable range where it 
was quite constant, and this range only was used in making experiments, the rate of 
descent being determined in addition during each observation. Guide wheels pre- 
vented the tilting of the gasometer during its descent, and the readings on the 
attached scale could therefore be relied upon. The pressure of the gas was 
determined by a manometer attached to the gasometer, and the pressure in the 
apparatus was similarly obtained. The volume of the gas emitted by the gasometer 
per second was then reduced to the pressure in the apparatus, and dividing by 
the flow area in the tubes, the required value of U in equation (9) was obtained. 

3. In order to understand more clearly the manner in which the values of 
A and X of equation (9) were determined, let us consider the following case. In 
fig. 3, CO represents a longitudinal section of the outer cylinder. DB is the 









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PRODUCED IN GASES BY RONTGEN RAYS. 



201 



inner cylinder having the insulated juncture at ky the part D of the cylinder being 
connected to an electrometer. 

The gas stream is supposed to flow from right to left in the figure, and hdmn 
is the beam of rays. DB being at zero potential, suppose that when the potential 
of CC is at a certain value the ions going towards DB move in paths parallel to the 
line ah in the upper half of the figure. An ion starting from any point to the 
left of ak would reach the part D and so influence the electrometer, but as all of the 
ions start from the beam of rays to the right of ak^ all of them reach B. If the 
potential of CC is diminished so that the inclination of the ionic paths becomes hk, 
ions from the outermost rim of hdmn will just begin to reach the part D. By a 
certain decrement in the potential of CC the paths of the ions can be made 
parallel to dk, so that ions will reach D from a volume whose section is represented 
by the triangle hdg^ the width of the beam of rays being hd. By a decrement in 
the potential of CC equal to the last one, the volume from which ions reach D is 
increased by a volume whose section is seen from the figure to be nearly a parallelo- 
gram of about twice the area of the triangle hdg. Another equal decrement in 
the potential increases the volume by almost the same amount as the last. As 
the potential is diminished further, the rate of increase of volume gradually 
diminishes. So if we represent the potentials used by abscissas and the volimies 
from which ions reach D by corresponding ordinates, we obtain a curve, fig. 4, 





















R 




















/ 








'^n 


p/a 


Cs, 






/ 








n 










/ 








<q 










J 


f 






' 










a 


/ 










^ 








/ 












£ 








1 


















/ 


















w 


/ 


















/ 




















/ 


















1 


















7 


/ 


















/ 


















u 


■yf 


















b 


C I 


i 


i 


z 


i 


f 









Fig. 4. 

whose inclination to the axis of abscissas, as the potentials are increased, at first 
gradually increases (RS of fig. 4), then assumes a constant value (ST) and finally 
diminishes (TU) as the curve ends in the axis of abscissas. The point U corre- 
voL. oxer.— A. 2 D 



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202 MR. J. ZELENY ON THE VELOCITY OP THE IONS 

spends to the inclination of the paths of the ions represented by hh of fig. 3, 
T corresponds to dk, and W to ek. As the paths change from ek to dk^ the 
diminution in the number of ions reaching D is equal to about twice the number 
that are getting to D in the latter case (W^ = 3 Td). If, therefore, the rate of 
diminution remained unchanged until ions just ceased to reach D, the change in 
potential required for this would be just a half of the change fi'om ek to dk or 
from dk to hk. Thus in the curve it is seen that by prolonging WT it reaches 
the axis at c, half way between 6 and d. This corresponds to a potential which 
would be required for an ion starting from c (fig. 3) the middle point of the beam 
of rays 6d, in order to have it just reach the juncture k in the inner cylinder. 
It is evident that the points T and U are not very sharply defined on an experi- 
mental curve, and hence cannot be determined as accurately as the point c, and so 
in practice the potential A of formula (9) has always been determined in this 
latter way. Evidently the value of X which is to be used with this value of 
A has to be measured from the middle of the beam of the rays where they cross 
the inner cylinder to the middle of the juncture k^ as aU ions reaching the middle 
point are drawn to D. The width of this juncture was only '05 centim. The 
width of the beam of rays was used as small as possible, and in most cases was 
•2 centim., this being a smaU part of the total distance X 

4. In considering the distribution of the ions between the two cylinders while the 
conduction is going on, it is seen from the lower part of fig. 3 that supposing the 
external tube to be positive, the negative ions starting from 5 will describe a path 
somewhat like sw, so that all of the negative ions will be confined to the space 
wmnts. Similarly the positive ions starting from m will describe the path mk^ 
and all of the positive ions will be confined to the space kmnts. In the space 
where these two overlap, i.e., omnts, both kinds of ions will be present and recom- 
bination will take place, the number of ions per cubic centim. diminishing, there- 
fore, as we go from 5m to o. 

The space ovnn will be occupied by negative ions alone, and oks by positive ions alone. 
wm will usually be shorter than ks^ because as a rule the negative ions travel the 
faster in the same electric field. 

5. Of the ions starting from m towards k all will not follow the path mi, but 
some, due to the motions assigned to them by the kinetic theory of gases, will 
difiuse to either side so that the distribution, along the path, of the ions which 
started from m will lie between the two dotted lines mr and mp. This effect will 
produce a distortion in such a curve as that shown in fig. 4, and to bring all of the 
ions to the part B of the inner cylinder will require a greater force than would be 
necessary if there were no diffusion. The effect of this disturbance upon the value 
of the ionic velocity obtained in the manner described is to give a result that is 
too small because the potential A obtained is too large. Moreover the amount of 
the diffusion depends upon the time required for the ions to travel between the two 



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PRODUCED IN GASES BY ItftNTGEN RAYS. 203 

cylinders so that if we obtain values of the ionic velocity, in the manner already 
described, these will be the larger and nearer to the true value the smaller the 
time that is required for the passage of the ions across. 

If this time were zero, then evidently all diffusion effects would disappear. 

6. The free charges that exist in the gas, where the ions of one sign predominate, 
tend to spread on account of the mutual repulsion of the charged carriers. 

This produces an eflFect similar to that of the diffiision just described. It increases 
with the time required for the ions to pass between the cylinders, but is less the 
smaller the density of the free charges, i.e., the weaker the Rontgen rays used and 
the narrower the beam of the rays. 

7. The presence of these free charges in the gas also has an influence upon the 
intensity of the electrostatic field between the two cylinders. To diminish this effect 
a sensitive electrometer was used in making the observations, as this allowed the 
employment of a weak radiation so that the charges in the gas were of a small 
density. 

While it is not possible to make an exact calculation of the magnitude of this effect 
because of the unsymmetrical distribution of the ions, an approximation to it can still 
be obtained. Knowing the capacity of the receiving system and the charge received 
in a given time, and knowing the approximate velocity of the ions in the electric field 
and the approximate space occupied by the firee charges, the density of these charges 
can be obtained roughly, and their effect upon the electrostatic field can be 
computed. 

Computations of this kind made from the observations used for final results 
showed that the largest value of this correction made a diminution in the electro- 
static field of less than 1 per cent. In some experiments where a large inner 
cylinder was used the intensity of the electric field employed was less, the ions 
moved slower, and the density of the fi^ee charges was therefore larger and in some 
instances the above correction was perhaps nearly 2 per cent. In all cases an 
increase in the strength of the field itself diminishes the percentage value of the 
correction, while the simultaneous diminution in the density of the free charges 
reduces it still further. 

8. The motion of these free charges through the gas also produces a motion of the 
gas itself, as the writer has previously shown.* The amount of this is, however, very 
smaU compared to the velocity of the ions, so that it cannot have an appreciable 
disturbing efiect upon the restdts of these experiments. 

9. In conduction produced by Rontgen rays there is a noticeable faU of potential at 
the electrodes which diminishes the electric intensity in the intermediate space. As 
determined by the writer, t for conduction in air between two plates 1*2 centims. 
apart, this amounted to about 2 per cent, of the total potential difference for the 

* J. Zeleny, ' Proc. Camb. Phil. Soc.,' vol. 10, Pt. L, p. 13. 
t J. Zbleny, ' Proc. Camb. Plul. Soc.,' vol. 10, Pt. I., p. 21. 

2D2 



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204 MR. J. ;ZeleNY on the velocity op the ions 

strength of rays used. For the much weaker radiation and the greater distance here 
used the correction does not perhaps exceed I'per cent. For gases other than air 
the effect has not been determined, and has been assumed to be no greater than with 
air. 

10. J. Perrin'*^ has shown that when the Rontgen rays impinge upon a metal 
surface the ionization in the gas near it is increased by an amount depending upon 
the nature of the metal and upon the state of its surface. M. G. SAGNAct and 
P. Langevin have shown since that this is due to a secondary radiation started at 
the metal surface by the Rontgen rays. It is possible that the ions so produced are 
of a different nature from those produced by the direct rays, but in the absence of 
any evidence to that effect the much more probable case is assumed that the two 
kinds are identical. 

The effect of the secondary rays, therefore, is to produce an uneven distribution ot 
the ions in the space exposed to the direct rays, and also to widen the ionized area 
near the metal surfaces. This makes more difficult the accurate determination of the 
potential A in equation (9), the tendency being to get it too large. J. Perrin found 
that the surface effect was by far the least for aluminium, what he calls the coefficient 
being '0 for aluminium in air as compared to '9 for gold in air. The effect is also 
very much dependent upon the cleanliness of the surface. It is thus seen that in the 
apparatus used this effect was made as small as possible by using unpolished 
aluminium as the material for those parts of the cylinders upon which the rays 
impinged. That the secondary rays did not produce an appreciable amount of 
ionization at a short distance to the side of the beam of the direct rays was shown 
by passing these rays near to the insulated juncture in Ihe inner cylinder while the 
gas in the tubes was at rest. No conductivity was observed to that part of the inner 
cylinder which was not exposed to the direct rays. 

Further experiments tried for the effect of the secondary rays by coating the inside 
of the aluminium cylinder on the apparatus with tin-foil will be described later among 
the observations for dry air. 

11. W. C. RontgenJ has shown that the air itself where it is exposed to the rays 
acts as a source of a weak secondary radiation. The writer is not aware of any 
experiments showing any conductivity produced by this radiation, but the experiment 
referred to in the last section, where a beam of rays near the juncture of the inner 
cylinder produced no appreciable conductivity on the other side, shows that in these 
experiments the effect may be disregarded. 

12. When D (fig. 3), the part of the inner cylinder joined to the electrometer, 
takes up a charge in the progress of an observation, the electric field in the vicinity 
of the juncture becomes slightly distorted, tending to lessen the number of ions 

* J. Perrin, 'Comptes Eendus/ vol. 124, p. 455. 
t M. G. Sagnac, 'Journal de Physique,' 1899, p. 65. 
X W. C. EONTGEN, * Wied. Ann.,' vol. 64, p. 18. 



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PEODUCED IN GASES BY RONTGEN RAYS. 205 

reaching D. As for each reading this effect starts from zero, the only influence of 
this upon a series of readings with difierent potentials is to diminish their values by 
small amounts nearly proportional to their size, thus having practically no effect upon 
the result obtained by projecting the curve as in fig. 4. 

13. The velocity of the ions is evidently dependent upon the pressure of the gas. 
In these experiments the variations in the pressure were but small, being due mainly 
to the variations of the barometer. No experiments have been carried out on the 
effect of pressure upon the velocity of the ions produced by Rontgen rays, but 
E. Rutherford* has shown that for the conduction produced by ultra-violet light 
the velocities of the ions in air are inversely as the pressure of the gas. This result 
will be used in these experiments to reduce all of the values of the velocities to the 
same pressure of 76 centims. of mercury. 

14. The effect of temperature upon the ionic velocity is not known, so that correc- 
tions for temperature could not be made. The temperature was, however, taken in 
all cases, so that if necessary the correction can be applied later on. 

15. In considering the various corrections above, it is seen that the effect of many 
of them is diminished or made negligible by using a narrow beam of weak rays, and 
by using unpolished aluminium for that part of the cylinders where the rays impinge. 
Those corrections which depend upon the time required for the ions to cross between 
the two cylinders could be made very small by sufficiently reducing the value of this 
time, but we are limited in doing so by the increase that is produced in the difficulty 
of measuring one of tlie required quantities. Resort must be had to finding the 
values of the ionic velocities for different times of crossing, and from these deriving 
the final results. 

An estimated correction of 2 per cent, will be made for those effects considered 
above, especially (7) and (9), which tend to make the result too small by an undeter- 
mined but small amount. 

§ 5. Changes made in Experimental Conditions. 

The apparatus used permits of several changes in the experimental conditions, 
which are a test of the accuracy of the method, and allow us to draw conclusions 
about the effects of some of the corrections previously noted. 

1. The velocity of the gas stream was varied by changing the weights on the 
gasometer. This necessitated a proportionate change in the value of the potential A 
of equation (9). The paths described by the ions are the same, but the time required 
for their passage between the two cylinders is changed. There are also changes in the 
amount of recombination of the ions and in the diffusion effect. The density of the 
free charges is changed, and so their effect upon the electric intensity is altered, and 
the spreading due to the mutual repulsion of the ions is also different. 

* E. KirniERFORD, * Proc. Camb. Phil. Soc.,* vol. 9, Pt. VIII., p. 4U. 



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206 . MR. J. ZELENY ON THE VELOCITY OP THE IONS 

2. The distance of the beam of rays from the insulated juncture in the inner 
cylinder was also changed. This likewise necessitated a change in the value of the 
potential A, but in the opposite sense. The paths of the ions are now quite 
different, and changes are also produced in all of the quantities mentioned in the 
preceding case. 

3. The intensity of the Rontgen rays was also varied. This produced alterations 
in the density of the free charges in the gas, and consequently in their effect upon the 
electric field between the cylinders and in the mutual repulsion of the ions. The 
amount of the recombination of the ions is also affected as well as the fall of potential 
at the electrodes. 

4. By changing the diameter of the internal cylinder complete changes are 
produced in the configuration of the forces, and of the motions of the ions. All the 
other changes can also be tried in conjunction with this one. 

5. The material of the inner surface of the outer cylinder was also altered to note 
the influence upon the result of increased ionization at the metal surface. 

6. In trying to find the effect of any of these changes upon the observed velocity 
the greatest difficulty met with is due to the smallness of the effects, and their conse- 
quent masking by the irregularities of individual observations caused by the difficulty 
of maintaining a uniform radiation for a length of time sufficient to cover a niunber 
of readings. Individual observations taken under the same conditions may vary 
among themselves by a number of per cent., so a smaU change in the result cannot 
be detected unless a large number of observations is made. 



§ 6. Method of Conducting the Experiments. 

The following procedure was followed in taking readings with the apparatus. The 
Crookes' tube and the lead slits were accurately adjusted, so that the beam of rays 
occupied the desired position, and the distance X of equation (9) was carefully 
measured. The cylinder AA' was connected to a chosen potential on the battery N. 
The electrometer quadrants, joined to the part B of the inner cylinder, were then 
disconnected from earth by means of the key K, and the zero reading was observed 
on the scale. The reading on the gasometer scale was also taken. At a definite 
time, observed on a chronometer, the valve at the gasometer was opened, so that the 
gas began to flow through the apparatus. After a short period, usually 10 seconds, 
sufficient to produce a steady state of flow in the apparatus, the primary of the 
induction coil was closed and the rays thus started. The rays were allowed to run 
for 30 seconds, and the primary of the coil was then broken, and the valve of the 
gasometer was also closed at a definite time. The electrometer reading was now 
taken, and the deflection produced was obtained. The key K was then closed, and 
the quadrants of the electrometer were connected to earth. From the reading on 



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PRODUCED IN GASES BY RONTGEN RAYS. 207 

the gasometer scale the volume emitted was obtained, and with the aid of the 
pressure readings which were taken the average velocity of the gas stream in the 
apparatus could be calculated. 

An interval of about three minutes was allowed as a rest for the tube, as this made 
it much more constant over a large number of readings. Iii the mean time, if 
necessary, gas was forced back from the gas bag into the gasometer. Guided by the 
previous electrometer deflection the potential of the outer cylinder was now changed, 
and the whole process repeated. In this way a number of readings were taken, such 
that the electrometer deflections ranged from some value down to near zero. These 
were taken in such an order that at first, say, a descending series of readings was 
obtained, and then immediately afterwards an ascending series. In this manner it is 
possible to detect any uniform changes which are taking place in the intensity of the 
rays, for in that case the two series of points would lie on curves of different 
inclinations. 

It was seen in § 5 that the time of passage of the ions from one cylinder to 
the other could be varied by changing the velocity of the gas stream, and also by 
changing the distance X. Both of these were employed in practice, and it was found 
that the values of the velocity obtained diminished as the time increased ; but they 
were practically the same for two different values of X if the velocity of the gas 
stream was changed in the same ratio, i.e., if the time of passage of the ions was 
the same. 

J, S. TowNSEND* has recently observed that the rate of diffusion of the ions 
depends upon the moisture in the gas. In these experiments the gases were used 
both dry and saturated with aqueous vapour, and it was found that the velocity was 
different in the two cases. 

For saturating a gas with aqueous vapour it was forced, in passing between the 
gasometer and the apparatus, to bubble through a water bottle and then to pass 
through a long horizontal tube half filled with water. After the gas had been passed 
several times back and forth between the gasometer and the gas bag, and before any 
readings were taken, the water bottle was cut out so as to avoid any unsteadiness in 
the pressure due to the bubbling. 

For drying a gas the above arrangement was replaced by one in which the gas 
had to pass through a long, horizontal glass tube, partly filled with concentrated 
sulphuric acid, and then through a large volimie of calcium chloride. In order to 
allow a suflBciently rapid stream with the small pressures used the calcium chloride 
was placed in a large, wide bottle, the gas entering above and leaving by a protected 
ftmnel-shaped tube near the bottom. It thus had to traverse a considerable length of 
calcium chloride, and on account of the large area of the bottle the velocity through 
it was smaU, 

♦ J, S. TowNSBND, 'Phil, Trans.,' A, vol. 193. 



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208 



MR. J. ZELENY ON THE VELOCITY OF THE IONS 



§ 7. Moist Air. 

The following is an example of a set of readings taken for the positive ions in air 
saturated with aqueous vapour. 

Letters refer to corresponding quantities in formula (9). 

Temperature = 14*5° C X = 2'60 centims. a = '50 centims. b = 

2 '555 centims. 
Width of beam of rays = '20 centim. Barometer =75*4 centims. 
Excess pressure inside gasometer =1*56 centims. of mercury. 

„ „ in apparatus = '59 centim. of mercury. 

20 cells = 42 "6 volts. 

Table I. — Moist Air. Positive Ions. 



Voltage of outer 
cylinder. • 


Electrometer deflection 


Descent of gasometer 


in 30 seconds. 


in 40 seconds. 


Cells. 


Divisions. 


Centims. 


+ 10 


145 


6-77 


+ 12 


105-5 


6-79 


+ 14 


68-5 


6-78 


+ 16 


29-5 


6-72 


+ 18 
+ 19 


12 


6-70 
6-83 


7 


+ 17 


19 


6-81 


+ 15 


52-5 


6-78 


+ 13 


87 


6-77 


+ 11 


128 


6-76 



In the middle of the observations the gasometer was refilled from the gas bag. 
The sectional area of the gasometer was 2904 sq. centims., and the area between the 
two cylinders was 19*73 sq. centims., so the average rate of descent of the gasometer 
above indicates an average velocity in the apparatus of 25*2 centims. per second, 
when corrected for the difference in pressure between the gasometer and the 
apparatus. 

The voltages and their corresponding deflections are represented graphically in 
curve I. of fig, 5. The set of readings here given, and most of those which are to 
follow as examples, have been selected from among the best obtained. 

It is seen that the curve at first approaches the axis of abscissas in nearly a straight 
line, but becomes convex when near to it. Had readings been taken for voltages 
smaller than those used, that part of the curve would have been concave to the axis 
of abscissas. 

It has been explained in § 4 (3), why there is a nearly straight portion in the 
curve, while the width of the beam of rays and the various causes tending to spread 



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PRODUCED IN GASES BY RONTGEN RAYS. 



209 



the ions make the lower end of the curve approach the axis at a less rapid rate. It 
was also shown that the point on the axis of abscissas, obtained by prolonging the 
straight portion of the curve, would correspond to the voltage required to make an 



















































/ 






















































/ 


























/ 


























} 










J. 


- /40 
















/ 










/ 


















r 
/ 










J 


-i£0 x^ 














/ 


f 










n 














1 


/ 










irZJ__ 


— 900 S 














/ 












^ 


- 'S 












/ 












7 _ 


-ao^ 






/ 












/ 




-^ 












1 












J' 




— AQ ^ 










/ 














1 














/ 














r 




~ 40 S 








i 


/ 














/ 




,-2 




i 


2 


1 
















/ 




— 20 


fr- 














/' 






-J 


F^ 


^ 














jr- 


^ 1 




.0 


?0 


1 




1 


6 

VoU 


/4 


! in 


1 




'ig. 


5. 




a 


^ 





ion, starting from the surface of the outer cylinder in the middle of the beam of rays, 
just reach the middle of the juncture in the inner cylinder. But with diffusion and 
the other causes acting to produce a spreading of the ions, it is evident that the 
inclination of the straight part itself is affected and the result changed. Corrections 
for this error can only be made in conjunction with those of some other effects, and 
that by experiment, by producing alterations in the amount of these effects, by 
changes in the time of passage of the ions across the space between the cylinders. 

The velocity obtained by the use of the voltage determined by the continuation of 
the straight part of the curve, as shown in the figure, will be called the ionic velocity 
for that determination, it being tmderstood that it is not implied thereby that the 
velocity changes with the time, but that this is only a step towards the final result. 
From the above curve, A is seen to be 177 cells, which is equal to 377 volts. 

Using equation (9), 

V = j — 5 — log<f~ f Tv = 5'118 ^^ — :^^ = 1'315 centims. per second, 

The pressure in the apparatus is 76 centims. of mercury 
VOL. cxcv.— A, 2 E 



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210 



MR. J. ZELENY ON THE VELOCITY OF THE IONS 



From equation (10) 



'*^ = i=ll = 'i*^^^"**- 



The following is a set of readings taken for negative ions in moist air. Unless 
otherwise mentioned the values of a, 6, and the width of the beam of rays will 
hereafter be taken the same as in the previous example. 

Temperature = 14*4° C. X = 6*42 eentims. Barometer = 74*7 eentims. 
Excess pressure inside gasometer = 1'54 eentims. of mercury. 

„ „ in apparatus = '59 centim. of mercury. 

8 cells =16-5 volts. 



Table II. — Moist Air. Negative Ions. 



Voltage of outer 
CTlinder. 


Electrometer deflection 


Descent of gasometer 


in 30 seconda. 


in 40 seconds. 


Cells. 


Division*. 


Centims. 


-4 


128 


605 


-5 


• 68-5 


5-95 


-5-4 


45 


5-94 


-6 


17-5 


5-92 


-5-6 


32-5 


5-89 


-5-2 


50 


5-90 


-6 


18 


602 


-5-4 


44-5 


5-99 


-5 


67-5 


5-96 


-4-4 


95 


5-90 


-7- 


2 


5-90 



The results are represented in Curve II. of fig. 5. 
U = 22*1 centims. per second. 



A = 127 volts. 



221 



= 1'39 centims. per second. 



The pressure in the apparatus =75*3 centims. 

The velocity reduced to 76 centims. pressure = 1'38 centims. per second. 

T = ^ = -29 second. 

The following is a summary of the results obtained for moist air for both the 
positive and negative ions. Each result was obtained from a series of observa- 
tions as indicated by the above examples. The results are reduced to 76 centims. 
pressure. 

Letters refer to quantities in equations (9) and (10). 



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PRODUCED IN GASES BY RONTGEN RAYS. 



211 







Tablk TTT. — Moiflt Air. Summary 


of Besult£ 


). 


















Ionic velocity. 


Reference 
number. 


X. 


U. 


A. 


T. 


Tempera- 
ture. 


Gas 
pressure. 
























Negative. 


Positive. 


1 


4-33 


20-6 


+ 19-2 


-21 


•c. 
15-3 


76-2 




1-28 


2 


4-18 


10-85 


+ 11-1 


-39 


15 


77-6 


— 


1-225 


3 


4-18 


111 


-10-1 


•38 


15 


77-6 


1-37 


— 


4 


418 


10-73 


- 9-8 


•39 


14-3 


76-5 


1-35 


— 


6 


4-18 


10-96 


+ 11-35 


-38 


14-3 


76-5 


— 


1-19 


6 


4-18 


25-0 


+ 23-4 


•17 


14-3 


76-8 


— 


1-32 


7 


418 


25-0 


-21-2 


-17 


14-3 


76-8 


1-46 


— 


8 


2-68 


11-3 


- 15-65 


-24 


14-6 


75-9 


1-38 


— 


9 


2-68 


11-3 


+ 17-5 


-24 


14-6 


75-9 


— 


1-23 


10 


2-68 


22-0 


+ 32-15 


-12 


14 


76-1 


— 


1-32 


11 


2-68 


221 


-29-1 


-12 


14 


76-1 


1-46 


— 


12 


8-41 


11-33 


+ 6-26 


-75 


14-4 


76-0 


— 


110 


13 


8-41 


11-23 


+ 6-03 


-75 


14-4 


76-0 


— 


113 


14 


8-41 


10-78 


- 5-2 


-78 


14-4 


76-0 


1-26 


— 


15 


8-41 


11-33 


- 5-41 


-74 


14-4 


76-0 


1-27 


— 


16 


8-41 


11-8 


- 5-51 


-72 


14-4 


760 


1-30 


— 


17 


8-41 


24-8 


- 10-95 


-34 


14-2 


76-2 


1-39 


— 


18 


' 8-41 


24-73 


+ 12-2 


-34 


14-2 


76-2 


— 


1-24 


19 


8-41 


24-8 


+ 12-5 


-34 


14-2 


76-2 


— 


1-22 


20 


8-41 


24-8 


-10-7 


-34 


14-2 


76-2 


1-42 


— 


21 


6-42 


10-67 


- 6-57 


-60 


13-5 


75-8 


1-30 


— 


22 


6-42 


10-7 


+ 7-26 


-60 


13-5 


75-8 


— 


1-175 


23 


6-42 


22-06 


+ 14-0 


-29 


14-4 


74-8 


— 


1-245 


24 


6-42 


22-06 


+ 14-15 


•29 


14-4 


74-8 


— 


1-23 


25 


6-42 


22-1 


-12-7 


-29 


14-4 


74-7 


1-38 


— 


26 


2-60 


11-1 


-15-7 


-29 


14-7 


74-7 


1-375 


— 


27 


2-60 


111 


+ 17-0 


-24 


14-7 


74-7 


— 


1-27 


28 


2-60 


25-2 


+ 37-7 


•10 


14-5 


75-4 


— 


1-315 


. 29 


2-60 


25-2 


-34-1 


•10 


14-6 


75-4 


1-466 


— 


30 


2-60 


12-6 


-17-8 


-21 


15 


75-5 


1-39 


— 


31 


2-60 


10-9 


-15-7 


-24 


15 


75-5 


1-37 


— 


32 


2-60 


13-1 


-18-3 


•20 


16 


75-7 


1-41 


— 


33 


2-60 


1316 


-17-8 


•20 


16 


75-7 


1-45 


-^ 




.7 



.a 



^ .3 

T. In aeconds. 

Fig. 6. 
2 E 2 



.£ 



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212 



MR. J. ZELENY ON THE VELOCITY OF THE IONS 



The results are represented iu fig. 6, where the velocities are represented as 
ordinates and the corresponding values of T as abscissas. It is seen that the 
velocities decrease with T, and nearly in a linear manner. Considerable variations 
are observed among the individual results, but it is believed that they are not 
greater than is to be expected from the nature and difficulties of the experiments. 

In § 4 it was seen that some of the corrections which act to give too small a value 
for the velocity diminish with T and disappear for T = 0. By drawing lines through 
the points in fig. 6 and prolonging them to the axis of ordinates where T = 0, we 
obtain the most probable values of the velocities. This gives for the negative ions 
1*48 centims. per second, and for the positive ions 1*34 centims. per second. 

In § 4 (15) it was stated that a correction of 2 per cent, would be made for dis- 
turbances not corrected by the above method. This gives for the final results for 
moist air the velocity in an electric field of 1 volt per centim. for the negative 
ions =1*51 centims. per second, and for the positive ions = 1'37 centims. per second 
at a temperature of about 14° C, and a pressure of 76 centims. of mercury. 

§8. Dry Air. 

The following set of readings was taken for the positive ions in dry air : — 

Temperature = 13'8° C. X = 2'60 centims. Barometer = 761 centims. 
Excess pressure in gasometer = 1 "6 centims, of mercury. 

„ „ apparatus = '45 „ „ 

14 cells = 29-0 volts. 

Table IV. — Dry Air. Positive Ions. 



Voltage of outer 
cylinder. 


Electrometer deflection 


Descent of gasometer 


in 30 seconds. 


in 40 seconds. 


Cells. 


Divi«ioiia 


Centims. 1 


+ 8 


117 


4-29 


+ 10 


60 


4-28 ! 


+ 12 


22 


4-26 


+ 14 


7 


4-26 


+ 11 


40 


4-22 


+ 9 


94 


4-2.5 


+ 7 


153 


4-25 


+ 10 


62 


4-21 ! 


+ 8 


123 


4-23 



These results are represented graphically in Curve I. of fig. 7. 
The corrected value of U is 15 '9 centims. per second. 

A = 24-8 volts. 

15-9 



Sov = 5-118 



2-60 X 24'8 



= 1*26 centims. per second, and when reduced to 



76 centims. pressure this becomes 1'27 centims. per second. 



rv 2-60 ,^ J 

T = — = -16 second. 



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PRODUCED IN GASES BY RONTGEN RAYS. 
The following set of readings was taken for the negative ions in dry air : — 

Temperature = 15'8° C. X = 2*60 centims. Barometer = 76 centims. 
Excess pressure in gasometer = TO centim. 
„ „ „ apparatus = '13 centim. 

7 cells =14-5 volts. 



213 




Fig. 7. 
Table V. — Diy Air. Negative Ions. 



Voltage of outer 
cylinder. 


Electrometer deflection 


Descent of gasometer 


in 30 seconds. 


in 40 seconds. 


Celli. 


DiTisions. 


Centims. 


-2 


141-5 


2-07 


-3 


87 


2-05 


-4 


28-5 


208 


-6 


6-5 


2-05 


-4-4 


18 


2-08 


-3-4 


63 


205 


-2-4 


116 


205 



The results are shown graphically in Curve IL of fig. 7. 

U corrected for pressure = 7 '64 centims. per second. 
A = 9-21 volts. 



Digitized by VnOOQ iC 



214 



V = 5-118 



MR J. ZELENY ON THE VELOCITY OF THE IONS 
7-64 



2-60 X 9-21 



— = 1*63 centima per second. 



T = - If? = -34 second. 

A summary of the results obtained for dry air for both the positive and the 
negative ions is given in Table VI. 

Table VI. — Dry Air. Summary of Results. 

















Ionic velocity. 


Beference 


X. 


U. 


A. 


T 


Tempera- 


Gas 






number. 


J- • 


ture. 


pressure. 




















Negative. 


Positive. 


I 


2-60 


7-64 


- 9-21 


34 


°c. 
15-8 


76-2 


1-63 




2 


2-60 


7-64 


+ 120 


34 


15-8 


76-2 





1-25 


3 


2-60 


7-64 


+ 12-2 


34 


16-3 


75-8 





1-23 


4 


2-60 


7-57 


- 9-31 


34 


16-3 


75-8 


1-60 





5 


2-60 


7-28 


- 915 


36 


12-2 


76-6 


1-58 





6 


2-60 


7-23 


+ 12-15 


36 


12-2 


76-6 





M8 


7 


2-60 


15-8 


- 18-3 


16 


13-8 


76-6 


1-71 





8 


2-60 


15-9 


+ 24-8 


16 


13-8 


76-6 


— 


1-27 


9 


2-60 


15-9 


-18.1 


16 


13-8 


7«-6 


1-74 





10 


2-60 


15-9 


-18-3 


16 


.13-8 


76-6 


1-725 





11 


2-60 


16-6 


- 18-65 


16 


14-6 


75-8 


1-75 





12 


2-60 


16-3 


- 18-22 


16 


14 


76-8 


1-78 





13 


2-60 


15-5 


+ 24-8 


17 


12-5 


76-7 





1-25 


14 


2-60 


15-5 


+ 23-7 


17 


10-7 


77-6 





1-31 


15 


2-60 


15-6 


-18-1 


17 


10-7 


77-6 


1-72 





16 


515 


8-65 


+ 7-76 


60 


11-4 


77-3 


— 


1-13 


17 


515 


8-62 


- 5-9 


60 


11-4 


77-3 


1-47 





18 


5-15 


15-6 


- 9-31 


33 


11-4 


77-6 


1-67 





19 


515 


15-7 


+ 12-6 


33 


11-4 


77-6 





1-26 


20 


515 


8-58 


+ 7-64 


•60 


11-7 


77-5 


_ 


114 


21 


5-15 


8^58 


- 5-88 


60 


11-7 


77-5 


1-48 






At No. 7 the drying apparatus was changed, and at No. 12 the Crookes tube 
was replaced by a new one. 

The results are represented in fig. 8, excluding the points marked by squares. 

The final values thus obtained for dry air when the 2 per cent, correction men- 
tioned in § 4 (15) has been added, give the velocity of the negative ions = 1 '87 centims. 
per second, and of the positive ions = 1*36 centims. per second. 

The temperature varied several degrees between the difierent observations, but 
was on the average about 13° C. 

Most of the tests to which the method used in these experiments was subjected 
by changes of experimental conditions, were tried with dry air. Among these was 
tried the effect of changes in the intensity of the rays. By interposing aluminium 
plates the rays were diminished so that the conductivities produced by them changed 



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PRODUCED IN GASES BY RONTGEN RAYS. 



215 



in the ratio of three to one, but no noticeable change in the result could be observed. 
During the course of all of the experiments the rays were not of the same intensity, 
for the Crookes' tube had to be replaced sevei'al times, but in aU cases without any 
marked effect upon the values obtained. It must be said, however, that rays of great 
intensity were never employed, the aim being always to have them as weak as 
possible for reasons previously stated. 















































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The most severe teat to which the method was subjected was a change in the 
dimensions of the inner cylinder. In the above experiments the diameter of this 
cylinder was 1 centim., and it was now exchanged for one having a diameter of 
2 '8 centims. The distance between the inner and the outer cylinders was thus 
diminished to nearly one-half of its former value. The electric field between the two 
became much more uniform, and the gas velocities for different points of a cross- 
section now varied in a different manner. In order to keep the other quantities the 
same, the small distance between the two cylinders necessitated the use of voltages 
only about one-quarter as large as those used in the former arrangement. This 
increased the difficulty of the measurements and also some of the corrections which 
must be applied to get the final result. The density of the free charges in the gas 
was greater because the ions moved slower, being in a weaker field, and the same fall 
of potential at the electrodes was a larger percentage of the total voltage. The 
width of the beam of rays used was '3 centim. 

The following is a summary of the results obtained : — 



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216 



MR. J. ZELENY ON THE VELOCITY OF THE IONS 





Table VII.— Dry 


■ Air. Summary 


for Large 


Inner Cylinder. 


















Ionic velocity. 


Beference 
number. 


X. 


U. 


A. 


T, 


Tempera- 
ture. 


Gas 
pressure. 
























Negative. 


Positive. 


1 


5-4 


10-7 


-1-90 


-50 


"C. 

irs 


75^9 


143 


_ 


2 


5-4 


12-6 


-2-23 


•43 


124 


759 


143 


— 


3 


5-4 


12-7 


+ 2-84 


•43 


124 


759 


— 


113 


4 


5-4 


10-4 


+ 2-35 


-52 


12^4 


75^9 


— 


110 


5 


3-77 


11-55 


+ 3-75 


-33 


15 


75^8 





112 


6 


3-77 


11-55 


-2-70 


•33 


15 


75^8 


1^555 





7 


3-77 


11-65 


-2-80 


-33 


15 


75^8 


151 





8 


511 


13-9 


-2-64 


-37 


154 


76^8 


1^43 





9 


5-11 


13-95 


+ 3-39 


•37 


15^4 


76^8 





1115 


10 


3-26 


13-8 


-406 


•24 


15^4 


76^8 


157 





11 


6-36 


8-83 


-1-48 


•72 


16 


77^2 


1315 





12 


6-35 


8-83 


+ 1-80 


-72 


16 


77^2 


— 


1-08 


13 


6-36 


14-05 


+ 2-80 


•45 


15^8 


76^0 


— 


1-09 


U 


6-35 


14-2 


-2-20 


•45 


15^8 


760 


140 





15 


2-63 


14-9 


-4-7 


•18 


15^8 


76-0 


165 





16 


2-63 


7-7 


+ 3-5 


•34 


163 


75^9 


— 


M4 


17 


2-63 


7-04 


r2-5 


•37 


163 


75^9 


1-465 





18 


2-63 


13-9 


-4-32 


•19 


16-3 


759 


167 





19 


2-63 


14-06 


+ 5-94 


•19 


16-3 


75^9 





1-23 



The results are represented in fig. 9. 

















































































































































































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^ 




















































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— 














































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PRODUCED IN GASES BY RONTGEN RAYS. 



217 



It is seen that in this case the values of the velocity change less rapidly as the 
values of T become large ; but for the smaller values of T the change is more rapid 
than it was when the smaller inner cylinder was used. The points on the curves are 
not advantageously distributed, and so do not allow of a very accurate projection of 
the lines to T = ; but from those drawn it is seen that the results are but slightly 
smaller than those obtained with the smaller inner cylinder. This is considered a 
good agreement even if it is left out of account that an additive correction is still to 
be made. 

An alteration which was tried to test the eflfect of surface ionization was a change 
in the material of the inner surface of the outer cylinder. The aluminium part DD' 
(fig. 2) of the outer cylinder was coated on its inner surface with a layer of tin-foil. 
The rays in penetrating the cylinder now had a tin instead of an aluminium surface 
in contact with the air. J. Perrin has shown (see § 4 (10) ) that what he calls the 
coefficient of the increased ionization at a metal surface is for tin in air '6 as against 
•0 for almninium in air. The effect varies with the state of the surface. In these 
experiments the aluminium surface used was an ordinary unpolished surface, while 
the tin surface used was that of bright tin-foil. It was thought that if an increase 
of the ionization near the metal surface has any marked effect upon the value of the 
velocity obtained, the difference should be observed by this new arrangement. 

The results obtained are given in Table VIII. , dry air being used as before. The 
smaller inner cylinder having a diameter of 1 centim. was used. 





Table VIII. 


— Dry Air. Summary for Tin Surfiice. 


















Ionic velocity. 


Beference 
number. 


X. 


U. 


A. 


T. 


Tempera- 
ture. 


Q&8 

pressure. 
























Negative. 


Positive. 


1 


6-22 


8-56 


- 606 


•6 


"C. 
139 


77-3 


142 




2 


6-22 


8-61 


+ 7-82 


•6 


139 


773 


— 


1105 


3 


6-22 


18-2 


+ 15-2 


•29 


14-4 


77^4 


— 


1-20 


4 


5-22 


18-2 


-10-8 


•29 


14-4 


77^4 


1^68 





5 


2-62 


17-3 


-19-6 


•15 


» 


77-2 


176 





6 


2-62 


18-5 


-21-2 


•14 


16-5 


77^2 


173 


— 


7 


2-62 


18-5 


+ 29-4 


•14 


16^5 


77^2 




1-25 



The points are plotted as squares on the curves in fig. 8, which represent corre- 
sponding values when the aluminium surface was used. It is seen that the points 
for the negative ions agree very well with the curve. The points for the positive 
ions are 2 to 3 per cent, below the values for the aluminium surface. Taking both 
results into consideration it was concluded, if the addition of a tin surface changed 
the values of the velocities by but such a small amount, that originally when the 

VOL. OXCV. — ^A. 2 F 



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218 



MR. J. ZELENY ON THE VELOCITY OF THE IONS 



aluminium surface was used, the eflfect of the surface ionization could not have been 
sufficient to produce any marked error in the results. 

The surface ionization also varies with the nature of the gas, but the values 
obtained by J. Permn for aluminium with the gases used in these experiments were 
in all cases much less than for tin in air. 



§ 9. Oxygen. 

The gas used in these experiments was the commercial oxygen obtained from a 
cylinder, which contained about 5 per cent, of impurities, mostly nitrogen. Since 
the size and nature of the apparatus prevented the employment of the most pure 
gases, it seemed advisable to use the cylinder gas. The density was changed but 
little by the presence of the impurities, and, so far as known, the velocity should 
therefore be but slightly aflfected. The drying of the gas and its saturation with 
aqueous vapour were carried out in the same manner as with air. The following set 
of readings was taken for the negative ions in oxygen saturated with aqueous 
vapour : — 



Temperature = 17-3° C. X : 
Excess pressure in gasometer 
„ „ apparatus 

8 cells = 16-3 volts. 



5*01 centims. 
1*54 centims. 
•55 centim. 



Barometer = 76*4 centims. 



Table IX. — Moist Oxygen. Negative Ions. 



Voltage of outer 


Electrometer deflection 


Descent of gasometer 


cylinder. 


in 30 seconds. 
DivisionB. 


in 40 seconds. 


Cells. 


Centims. 


-3 


194-5 


6-06 


-4 


157 


511 


-5 


106-5 


5-02 


-6 


41 


511 


-6-6 


17 


5-07 


-5-6 


67 


5-08 


-5-2 


93-5 


509 


-4-6 


125-5 


5-03 



The results are shown in Curve I. of fig. 10. 

The corrected value of U = 1883 centims. per second 

A = 13-55 volts. 

18-83 



V = 5-118 



= 1*413 centims. per second, and when reduced to 



T = 



501 X 13-55 
76 centims. pressure this becomes 1'43 centims. per second, 

5-01 



18-83 



= '27 second, 



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PRODUCED 


IN 


GASES 


BY RONTGEN RAYS. 


















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219 



The following is a set of readings taken for the positive ions in oxygen saturated 
with aqueous vapour : — 

Temperature = 15*6° C. X = 5'01 centims. Barometer = 76-5 centims. 
Excess pressure in gasometer = '44 centims. 

„ „ apparatus = '16 „ 

6 ceUs = 12-35 volts. 

Table X. — Moist Oxygen. Positive Ions. 



Voltage of oater 


Electrometer deflection 


Descent of gasometer 


cylinder. 


in 30 seconds. 


in 40 seconds. 


Cells. 


Diviaions. 


CentiniB. 


+ 2 


US 


2-30 


+ S 


85-6 


2-31 


+ 3-6 


20-6 


2-29 


+ 3-2 


41 


2-29 


+ 2-8 


73-6 


2-27 


+ 2-4 


112 


2-25 


+ 2-2 


127 


2-26 



The results are represented by Curve 11. of fig. 10. 

2 F 2 



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220 



MR. J. ZELENY ON THE VELOCITY OF THE IONS 



The corrected value of U = 8 "42 centims. per second. 



A = 7-42 volts. 



V = 5-118 



8-42 



= 1'16 centims. per second, and when reduced to 



5-01 X 7-42 
76 centims. pressure this becomes 1'17 centims. per second. 



T = —r^ = '6 second. 

o'4^ 



A summary of the results thus obtained for moist oxygen for both the positive and 
the negative ions is given in Table XI. 

Table XI. — Moist Oxygen. Summary of Results. 



Beference 
number. 


X. 


U. 


A. 


T. 


Tempera- 
ture. 


Gas 
pressure. 


Ionic velocity. 


Negative. 


Positive. 


1 
2 
3 
4 
5 
6 
7 
8 
9 
10 


6-89 
6-89 
6-89 
6-89 
5-01 
5-01 
501 
5-01 
3-03 
3-03 


9-41 
9-38 
16-8 
16-8 
8-42 
8-68 
18-8 
18-6 
20-5 
20-6 


+ 604 

- 5-38 

- 8-80 
+ 10-5 
+ 7-42 

- 6-69 
- 13-55 
+ 15-7 
+ 27-3 

- 24-25 


•73 

•73 

•41 

•41 

•6 

•58 

•27 

•27 

•15 

•15 


"C. 
15^4 
154 
156 
15-5 
15-6 
15-6 
17-3 
17-3 
15-4 
15-4 


76-8 
76^8 
773 
773 
76^7 
76^7 
769 
769 
76-9 
76-9 


1-31 
1-44 

1-34 
1-43 

1^46 


117 



fil 
117 

1-23 

1-29 





The results are shown graphically in fig. 11. 




.6 



.5 



T in seconds. 

Fig. 11. 



.3 



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PRODUCED IN GASES BY RONTGEN RAYS. 



221 



The correction mentioned in § 4 (15), which is to be applied to the values indicated 
at T = has in this case been reduced to 1 per cent., 1 per cent, being allowed for 
an increase in the velocity due to a diminution of density caused by the impurities in 
the gas. The corrected value thus obtained for the velocity in moist oxygen is for 
the negative ions = 1'52 centims. per second, and for the positive ions = 1*29 
centims. per second, at a pressure of 76 centims. and at a temperature of about 
16° C. 

The following is a summary of the results obtained for the positive and negative 
ions in dry oxygen : — 

Table XII. — Dry Oxygen. Summary of Results. 



Beference 
number. 


X. 


U. 


A. 


T. 


Tempera- 
ture. 


Gas 
pressure. 


Ionic velocity. 


Negative. 


Positive. 


1 
2 
3 
4 
6 
6 
7 
8 
9 

10 
11 
12 
13 
14 


2-73 
2-73 
2-73 
2-73 
2-73 
3-89 
3-89 
3-89 
3-89 
3-89 
6-89 
6-89 
6-89 
6-89 


13-9 
13-8 
13-3 
13-3 
13-3 
16-65 
17-7 
16-9 
7-97 
8-46 
8-92 
9-07 
17-35 
17-4 


+ 20-2 
- 15-37 
+ 19-5 
-14-7 
+ 19-5 
-13-6 
+ 17-7 
-13-4 

- 6-82 
+ 9-45 
+ 5-64 

- 4-63 

- 8-06 
+ 10-33 


-20 
-20 
-21 
•21 
-21 
-23 
-22 
-23 
•49 
•46 
-77 
-76 
-40 
-40 


"0. 
20-3 
20-3 
19-4 
19-4 
19-4 
16 
15-6 
16-6 
16 
16 
15-2 
15-2 
15-8 
15-8 


77-3 

77-3 

77-4 

77-4 

77-4 

78 

78 

78 

76-8 

76^8 

77-1 

77-1 

77-1 

77-1 


1-71 

1-72 

1-66 

1-71 
1-56 

fsi 

1-63 


1-31 
1-31 
1-30 
1-35 

1-19 
1-19 

1-27 



The results are represented graphically in fig. 12. 

When, as in the case of moist oxygen, a 1 per cent, additive correction is applied 
to the values indicated in the figure by T = 0, the final result for the velocity in dry 
oxygen is for the negative ions = 1'80 centims. per second, and for the positive ions 
= 1'36 centims. per second for a pressure of 76 centims. and a temperature of about 
17^ C. 



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222 



MR. J. ZELENY ON THE VELOCITY OF THE IONS 















— 1 






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Fig. 12. 



§ 10. Carbonic Acid. 

The gas used was taken from a cylinder of liquid carbonic acid. The small amount 
of impurities in this does not produce any marked change in the density of the gas, 
and is assumed to be without noticeable effect upon the ionic velocities. As examples 
of the readings taken, the following two sets are given for carbonic acid gas saturated 
with aqueous vapour : — 



Temperature = 16'3° C. X = 3-02 centims. 
Excess pressure in gasometer = '44 centim. 

„ „ „ apparatus = '21 „ 

10 cells = 20-6 volts. 



Barometer = 75*4 centims. 



Table XIII, — Moist Carbonic Acid. 


Negative Ions. 


Voltage of outer 
cylinder. 


Electrometer deflection 


Descent of gasometer 


in 30 seconds. 


in 40 seconds. 


Celb. 


DiTisions. 


Centims. 


- 6 


136 


2-34 


- 7 


106-5 


2-33 


- 8 


76 


2-33 


- 9 


41-5 


2.30 


-10 


17-5 


2-33 


- 9-4 


28 


2-29 


- 8-4 


57 


2-28 


- 7-4 


91-5 


2-27 



Digitized by VjOOQ IC 



PRODUCED IN GASES BY RONTGEN RAYS. 
The results are shown in Curve I. of fig. 13. 



2'23 



lao 




VolCcL^e in ceils. 
Fig. 13. 

The corrected value of U = 8*52 centims. per second. 



A = 21'1 volts. 

'302 n4 ~ ^^^ centim. per second, which, reduced to 76 



V = 5-118: 



centims. pressure, becomes '679 centim. per second. 



m 302 „^ J 

T = T7^ = -36 second. 



Table XIV. — Moist Carbonic Acid. Positive Ions. 



Voltage of outer 
cylinder. 


Electrometer deflection 


Descent of gasometer 


in 30 seconds. 


in 40 seconds. 


OelU. 


Divisions. 


Centims. 


+ 2 


176-5 


2-32 


+ 3 


170 


2-31 


+ 6 


106 


2-33 


+ 7 


77-5 


2-32 


1 +8 


45 


2-32 


+ 9 


17 


2-32 


+ 8-4 


28 


2-26 


+ 7-4 


62 


2-25 


+ 6-4 

i 


97-5 


2-27 



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224 



MR. J. ZELEITS ON THE VELOCITY OF THE IONS 



These results are represented in curve II. of fig. 13. 
The corrected value of U = 8 "48 centims. per second. 



A = 19-16 volts. 



8-48 



.= "749 centim. per second, which reduced to 76 centims. 



" - ^'^^^ 302 X 1916 

pressure becomes "745 centim. per second. 

3*02 
T = r-^ = -36 second. 

o.4o 



A summary of the results for moist carbonic acid is given in Table XV. 
Table XV. — Moist Carbonic Acid. Summary of Results. 



Reference 
number. 


X. 


U. 


A. 


T. 


Tempera- 
ture. 


Gas 
pressure. 


Ionic velocity. 


Negative. 


Positive. 


1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 


302 
3-02 
302 
3-02 
6-07 
6.07 
607 
607 
6-07 
6-07 
6-07 


8-48 
8-52 

16-6 

16-75 
9-74 
9-97 

19-4 

181 
8-63 

1305 

12-9 


+ 1916 
-21-1 
-39-2 
+ 35-9 
-12-3 
+ 11 
+ 21-5 
-22-14 
+ 9-95 
+ 14-1 
- 15-85 


-36 
-36 
•18 
■18 
-62 
-61 
•31 
•33 
•70 
•47 
•47 


°C. 

163 

163 

166 

16-6 

16^9 

16-9 

17^1 

171 

17^1 

176 

176 


756 

75-6 

76-2 

76^2 

75 

75 

75-5 

75-5 

75 

75^1 

75^1 


•679 
•717 

•658 
•685 
•678 


•745 

•791 

•755 
-756 

•722 
•772 



















































































































^^ 




^ 
































P 


73/ 


f/V 


e. 









— 




"^ 


■^ 




















. 














» 


. . 




-— 


-* 




X 


































^^ 


^ 


— 


— ■ 




"*" 












X 






















^^ 




^ 


— ■ 




































, 




— 


^ 


' — 


"^ 


































ft- 


^^ 




— - 





"^ 






































a— 


...-. 


— 




— 










Ni 


J?^ 


th 


e. 






























"^ 












































































































































































•1 
































































• i 


7 






X 


\ 






.i 


1 






4 


^ 






.« 








./ 


» 






mi 


f 










^i 



i 



4^ 



J 



T in seconds. 

Fig. 14. 



The results are represented in fig. 14, from which it is seen that the values 
corresponding to T = 0, when corrected similarly to those of moist oxygen, give as 
the velocity in moist carbonic acid, for the negative ions, 75 centim. per second, and 



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225 



for the positive ions, '825 centim. per second, for a pressure of 76 centims. and a 
temperature of about 17° C. 

A summary of the results obtained for dry carbonic acid is given in Table XVI. 

Table XVI. — Dry Carbonic Acid. Summary of Results. 



Beference 
number. 


X. 


U. 


A. 


T. 


Tempera- 
ture. 


Gas 
pressure. 


Ionic Velocity. 


Negative. 


Positive. 


1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 


607 
607 
607 
607 
308 
308 
308 
308 
601 
6-01 
6-01 
601 


8-61 
8-63 
16-7 
171 
17-3 
17-3 
8-25 
8-53 
8-63 
8-53 
12-8 
12-85 


- 915 
+ 9-59 
+ 19-07 
- 18-05 
-36-2 
+ 38-4 
+ 18-96 
-18-18 

- 9-53 
+ 9-64 
+ 14-76 
- 1404 


•71 
•71 
•36 
•36 
•18 
•18 
•37 
•36 
•70 
•71 
•47 
•47 


-c. 

17-5 
17-5 
175 
175 
18-3 
183 
17-2 
17-2 
173 
173 
17-5 
17-5 


75-4 
75-4 
75-8 
75-8 
75-8 
75-8 
75-7 
75-7 
75-7 
75^7 
75-9 
75-9 


-787 

•796 
•793 

•781 
•770 

•777 


•752 
•737 

•747 
•725 

•753 
•738 



The results are represented in fig. 15. 

The velocities appear to vary but little with T. 

The values for the positive velocity being comparatively large for the highest value 
of T, make it difficult to draw the line through the positive points, and the inclination 
of the one through the negative points has been used as a guide for drawing the one 
shown. The value thus found, when corrected, gives the velocity in dry carbonic acid 
for the negative ions as '81 centim. per second, and for the positive ions as 76 centim. 
per second for a pressure of 76 centims. and a temperature of 17*5° C. 































































i 






























N€ 




S/K 


?. 












=sss 








— 




^ 


— 


-<t 








= 












°-" 








_a 
















^^^ 








^ 




— 


= 






























X 


-F\ 


^Si 


Wi 


f. 






















y% 




























































































































J 


























































































































































































. d 


f — 






.€ 


\ 






4 


1 






,4 


^ 






•% 








1 


\ — 






•i 


t 






s 





VOL. CXCV. — A. 



T in seconds. 

Fig. 15. 
2 G 



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226 



MR. J. ZELENY ON THE VELOCITY OF THE IONS 



§ 11. Hydrogen. 

The gas was prepared from pure zinc and hydrochloric acid, and bubbled through 
three bottles of strong caustic potash and potassium permanganate to free it from the 
acid and other impurities. Great difl&culty was experienced in maintaining the gas 
sufficiently pure on standing, because of the large surface of rubber exposed in the 
gas bag and in the connecting tubes of the apparatus. The density of hydrogen 
being so small compared to that of air, a small amount of the latter produces a large 
change in the density of the gas, and it was found that the ionic velocities were 
greatly affected thereby. 

The following plan was finally adopted as the most practicable under the circum- 
stances : — 

The forenoon of a day was spent in the preparation of fresh hydrogen, this length 
of time being required to generate the large quantity necessary for use and for 
washing the more impure hydrogen out of the apparatus. Beginning early in the 
afternoon, readings were taken as rapidly as possible until after midnight, thus giving 
about eleven hours of continuous observations. The density of the gas was then 
determined by weighing a 600 cubic centims. flask filled first with dry air and then 
with dry gas from the gasometer. Since 1 per cent, of air in the gas made a difference 
of over 6 milligrams in the weight, this permitted a sufficiently accurate determination 
of the amount of the air impurity. A test was made by the eudiometer method, 
which showed that the impurity was practically all air. 

The width of the beam of rays used was '3 centim., as the conductivity was much 
less with the hydrogen than in the other cases. 

The following is a set of readings taken for the negative ions in dry hydrogen : — 

Temperature = 20° C. X = 2*95 centims. Barometer = 76*15 centims. 
Excess pressure in gasometer = '90 centim. 
„ ,. apparatus = 'SG centim. 

5 cells = 10*5 volts. 

Table XVII.* — Dry Hydrogen. Negative Ions. 



Voltage of outer 
cylinder. 


Electrometer deflection 


Descent of gasometer 


in 30 seconds. 


in 40 seconds. 


Cells. 


Diviiiouii. 


Centinis. 


-2-6 


12-5 


9-59 


-3 


9-7 


9-52 


-3-4 


6-5 


9-31 


-3-8 


3-8 


9-44 


-3-6 


6 


9-46 


-3-2 


. 8-8 


9-50 


-2-8 


11-2 


9-38 


-2-4 


14-2 


9-40 



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227 



The results are shown graphically in curve II. of fig. 16. 
The corrected value of 17= 35*0 centims. per second. 



A = 9-04 volts. 



V = 5-118 



35 



2-95 X 9-04 
pressure becomes 676 centims. per second. 

9-Q5 

T = ^ - = -084 second. 
The gas in this case contained 3 '4 per cent, of air. 



= 672 centims. per second, which reduced to 76 centims. 







L 




f 








n 

J 




/ 


lA 






7 




f 
J 


F^ 






t 




/ 


lO 




7 




y 




% 




L 




/ 




A 




1 




/ 




^ 




t 




7 




^»^ 




'4 


M 


/ 




O 




t 




t 








f 
-J 


7 










7 


A 






4± ^ 




( 


7 






i^"' 




7 


f 
J 










t 


/ 








-lV 


: : ( ^. i 


u 









4t. 



VoLta^ in ceUs. 

Fig. 16. 

The following set of readings was taken for the positive ions in hydrogen saturated 
with aqueous vapour : — 

Temperature = 20° C. X = 2*95 centims. Barometer = 767 centima 
Excess pressure in gasometer = '78 centim. 
„ „ apparatus = '35 centim. 

9 cells =18-5 volts. 



2 G 2 



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228 



MR. J. ZELENY ON THE VELOCITY OF THE IONS 



Table XVIII. — Moist Hydrogen. Positive Ions. 





Voltage of outer 


Electrometer deflection 


Descent of gasometer 




cylinder. 


in 30 seconds. 


in 40 seconds. 




Cell.. 


DiTisiona. 


Centims. 




+ 6-6 


5-25 


10-38 




+ 6 


7-75 


10-23 




+ 5-4 


10-25 


10-18 




+ 4-6 


14 


10-38 




+ 5-4 


10-75 


10-25 




+ 6 


7-75 


10-18 




+ 6-6 


5 


10-16 



The results are shown in curve I. of fig. 16. 

The corrected value of U = 43 '3 centims. per second. 

A = 15-9 volts. 
43-3 



i; = 5-118 



<^n^ i crn = 473 centims. per second, which reduced to 76 centims. 
2-95 X 15-9 ^ 

pressure, = 4'80 centims. per second. 

2*95 
T = ^ = '068 second. 

Besides the water vapour, the gas in this case contained 1 "5 per cent, of air. 
A summary of the results obtained with dry hydrogen containing 3*4 per cent, of 
air is given in Table XIX. On account of the smaller electrometer readings the 

Table XIX. — Dry Hydrogen. Summary of Results. 



Beference 
number. 


X. 


U. 


A. 


T. 


Tempera- 
ture. 


Gas 
pressure. 


Ionic velocity. 


Negative. 


Positive. 


1 
2 

3 
4 
5 
fi 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 


2-95 
2-95 
2-95 
2-95 
2-95 
2-95 
2-95 
2-95 
2-95 
2-95 
2-95 
2-95 
2-95 
2-95 
2-95 
2-95 


21-2 
21-6 
21-8 
21-9 
21-6 
21-4 
21-4 
21-4 
35-4 
35-3 
35-0 
35-0 
34-9 
34-7 
25-7 
25-7 


+ 6-65 
+ 7-73 
+ 7-14 
+ 7-37 

- 5-79 

- 5-75 

- 5-85 
+ 7-14 
+ 10-35 
+ 10-79 

- 9-03 

- 9-24 

- 8-90 
+ 11-10 
+ 8-08 
+ 7-90 


-083 
-083 
-084 
•084 
-084 
-085 
-115 
-115 


•u. 

21-4 

21-4 

21-4 

21-4 

21-4 

21-4 

21-4 

21-4 

20 

20 

20 

20 

20 

20 

20 

20 


76-3 

76-3 ' 

76-3 ' 

76-3 

76-3 

76-3 

76-3 

76-3 

76-5 

76-5 

76-5 

76-5 

76-5 

76-5 

76 4 

76-4 


6-52 
6-49 
6-39 

6-77 

6-eo 

6-84 


5-56 
5-15 
5-33 
5-18 

5-24 
5-97 
5-71 

5-45 
5-55 
5-67 



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229 



determinations for hydrogen are less accurate, and so results were obtained for small 
values of T, only because of their greater importance, and in order to expedite the 
readings. 

The results are represented by I. and II. of fig. 17. 

The lines projected to T = indicate for the uncorrected velocity of the negative 
ions 7 '3 centims. per second, and for the positive ions 6*2 centims. per second when 
under a pressure of 76 centims., and at a temperature of about 20° C. These values 
are for dry hydrogen containing 3*4 per cent, of air. The correction for the presence 
of the air can be found approximately by finding the value of the velocity in a gas 
having a larger percentage of air. 



z 

































































Ni 


5^ 


th 


e. 


y 


X' 






























/. 


.^ 


y 






7. 


























O J 


/ 


o 
































y 


^ 












i 




























i 


°« 


iCi 


ve. 


y 


«0 






























— 3r 


y 


X' 




« 2j, 


























X 


X 


^ 








• t^ 














M 












y 




X 


















f 


^ 












» 


n. 










•5 




w 


-^ 




o 


X 


^ 


























^ 


''^ 




^ 




























^ 




-^ 


^* 




X 














n, 














^ 






































»l 


\ 


























• 










♦-2 
























x> 




















































• 









































T in seconds. 

Fig. 17. 



./ 



The following are a number of results obtained with dry hydrogen which contained 
14*4 per cent, of air : — 

Table XX. — Dry Hydrogen with 14*4 per cent, of Air. 



Reference 
number. 


X. 


U. 


A. 


T. 


Tempera- 
ture. 


Gas 
pr/)ssure. 


Ionic velocity. 


Nogative. 


Positive. 


1 
2 
3 
4 


2-95 
2-95 
2-95 
2-95 


20-4 
20-2 
18-6 
18-6 


+ 9-62 
-8-28 
-7-59 
+ 8-90 


•15 
•15 
•16 
•16 


"C. 
21-2 
21-2 
■ 22 
22 


76-3 
76-3 
76-4 
76-4 


4-25 

4-27 


3-70 
3-65 



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230 



MR. J. ZELENY ON THE VELOCITY OF THE IONS 



These results are represented at III. in fig. 17. 

By finding the difference between these points and the values in the curves above 

them corresponding to the same value of T, the diminution in the velocity is obtained 

that is produced by the addition of 14*4 — 3 4 = 11 per cent, of air. Assuming that 

up to this point the diminution in the ionic velocity is proportional to the amount of 

air present in the gas, the velocity in pure hydrogen is found by adding to the value 

34 
obtained when 3 '4 per cent, of air was present tj part of the diminution observed as 

due to 11 per cent, of air. From the above results this correction is found to be '65 
for the negative ions and '50 for the positive ions. Disregarding any minor correc- 
tions, the final result for pure dry hydrogen is thus found to be 7 '9 5 centims. per 
second for the negative ions, and 6*70 centims. per second for the positive ions at a 
pressure of 76 centims., and at a temperature of about 20° C. 

A summary of the results obtained with hydrogen saturated with aqueous vaj)our, 
and containing 1 '5 per cent, of air, is given in Table XXI. 





. Table XXI.- 


—Moist Hydrogen. Summary of Results. 


















Ionic velocity. 


Reference 
number. 


X. 


U. 


A. 


T. 


Tempera- 
ture. 


Gas 






pressure. 


















Negative. 


Positive. I 


1 


2-95 


43-9 


-U-6 


•067 


20 


771 


526 




2 


2-95 


43-7 


-15-0 


•067 1 


20 


771 


510 


— 


3 


2-95 


43-7 


+ 15-5 


•067 


20 


77-1 


— 


4-97 


4 


2-95 


43-3 


+ 15-9 


•068 


20 


771 


— 


4-80 


5 


2-95 


431 


-150 


•069 


20 


771 


503 


— 


6 


2-95 


23-8 


- 8-43 


•12 


19^8 


.76^9 


4^98 


— 


7 


2-95 


23-5 


- 8-76 


•13 


19^8 


76^9 


4-72 


— 


8 


2-95 


23-4 


- 8-64 


•13 


19^8 


76^9 


4^77 


— 


9 


2-95 


23-3 


+ 8-77 


•13 


19^8 


76-9 





4-68 


10 


2-95 23-3 


+ 8-69 


•13 


198 


76^9 





4-73 


11 


2-95 I 34-2 


+ 13-3 


•087 


19^8 


77 





4-53 


12 


2-95 


341 


+ 13-8 


•087 


19^8 


77 





4-37 i 


13 


2-95 


34 


-12-4 


•087 


198 


77 


4-82 


— 


14 


2-98 


19-8 


- 8-74 


•15 


204 


76-7 


3-93 


— 


15 


2-98 


19-7 


+ 8-97 


•15 


20-7 


76^9 


■~~" 


382 



air 



The results 1 to 13 are represented by IV. of fig. 17. 

The resuJt^ 14 and 15 were obtained with moist hydrogen containing 8 per cent, of 
ThesQ itwp were selected out of a number of results of which they represent 
about the ,averag€ values. They are shown by V. of fig. 17, and by means of them 
the correction for the air present in the above experiments was made in the same 
manner as with dry hydrogen. The points IV. in the figure are so scattered that the 
inclination of the lines drawn through them had to be estimated mainly by 
comparison with those for dry hydrogen, remembering that with the smaller 



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231 



velocities here obtained the inclination would be somewhat less. The final values 
thus obtained for hydrogen saturated with aqueous vapour when corrected for the air 
present give for the velocity of the negative ions 5 '6 centims. per ^second, and for the 
positive ions 5*3 centims. per second at a pressure of 76 centims., and a temperature 
of 20° C. 

§ 12. Eemarks on the Experiments. 

The changes in the values obtained for the velocity with changes of T are 
observed to be greater for those cases where the ionic velocities are higher. 
With dry and moist carbonic acid, however, the inclination of the ciu*ves is some- 
what different for nearly equal values of the velocities. In some instances, where 
the set of points for either the positive or the negative ions did not allow of a 
suflSciently accurate estimate of the inclination of the line to be drawn through them, 
the line through the other set of points was used as a guida 

The presence of water vapour diminished the velocity of the negative ions in all of 
the gases, while in carbonic acid the velocity of the positive ions was at the same 
time considerably increased. It seems most probable that these changes are due to 
some effect upon the size of the ions, and it is possible that a few molecules of the 
aqueous vapour collect upon the negative ions. It is interesting to note in this con- 
nection the recent results of C. T. R. Wilson,* showing that in supersaturated air 
the water condenses more readily upon the negatively charged ions. 

While in most cases the readings indicate a greater accuracy, it is believed that 
the maximum error in any determination is less than five per cent. For convenience, 
all of the values obtained are here collected in one table, the results being given in 
centims. per second both for a field of one volt per centim. and for a field of one 
electrostatic unit per centim. 

Table XXII. — Ionic Velocities. 



Gas. 


Velocities in centims. 

per second in a field of 

1 volt per centim. 


Velocities in centims. 

per second in a field of 

1 E.S.U. per centim. 


Eatio 

of 
Negative 

to 
Positive. 


Tempera- 
»ture. 


Positive 
ions. 


Negative 
ions. 


Positive 
ions. 


Negative 
ions. 


Air, dry 

„ .moist 

Oxygen, dry .... 

„ moist . . . 

Carbonic acid, dry . . 

„ „ moist . 

Hydrogen, dry . . . 

„ moist . . 


1-36 
1-37 
1-36 
1-29 
•76 
•82 
670 
5-30 


1-87 

r5i 

1-80 
1^52 
•81 
•75 
7^95 
5-60 


408 
411 
408 
387 
228 
246 
2010 
1590 


561 
453 
540 
456 
243 
225 
2385 
1680 


1-375 
MO 
1-32 
1^18 
107 
•915 
119 
1-05 


"C. 
13-5 
14 
17 
16 
175 
17 
20 
20 



* C. T. R. Wilson, *Phil. Trans.,' A, vol. 193, p. 289, 1899. 



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232 MK. J. ZELENY ON THE VELOCITY OF THE IONS 

It is seen that the value of the velocity is greater for the negative ions in all cases 
except for moist carbonic acid. In comparing the values for the diflferent gases, the 
temperature at which the observations were taken must be taken into consideration. 

At the time the writer* determined the ratio of these velocities, the influence of 
moisture being unknown, the gases used were not dried, and so the values obtained 
were between those given above for the dry and the moist gases. Of the gases used 
in the former experiments, which were not used in these, the ammonia gas used had 
been passed through two long tubes of calcium oxide, the acetylene gas had been 
passed through a long tube of calcium carbide and the nitrogen monoxide was used 
directly from a cylinder. 

The results obtained by E. RuTHERFORDt for the sum of the velocities of the ions 
produced by Rontgen rays are for : — 

Air = 3'2 centims. per second. 
Oxygen =2*8 centims. per second. 
Carbonic acid = 2*15 centims. per second. 
Hydrogen = 10*4 centims. per second. 

It is not stated whether the gases were dried, but the value for air agrees with 
that given above for the sum of the velocities in dry air, while the values for oxygen 
and hydrogen agree with the values for the moist gases. The value for carbonic acid 
is nearly 40 per cent, larger than that obtained here. It is of interest to compare 
the velocities of the ions produced by Rontgen rays with those of the ions produced 
by the action of ultra-violet light and in the discharge from points, as they show a 
close similarity. 

For conduction produced by ultra-violet light, E. Rutherford:|; obtained with 
dry gases for the velocity of the negative ions in — 

Air = 1'4 centims. per second. 
Hydrogen = 3*9 centims. per second. 
Carbonic acid = '78 centim. per second. 

The value for carbonic acid is quite near to that obtained above, but the other two 
are considerably smaller. 

A. P. Chattock§ obtained for the velocities of the ions in dry air in the case of 
discharge from points — 

413 centims. per second for the positive ions, and 

540 „ „ „ negative ions for a field of one electrostatic unit. 

* J. Zeleny, *Phil. Mag.,' July, 1898. 

t E. Rutherford, ' Phil. Mag.,' November, 1897. 

t E. RuTHERiX)iU), ' Proc. Camb. Phil. Soc.,' vol. 9, Pt. VIII. 

§ A. P. Chattock, *Phil. Mag.,' November, 1899. 



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PRODUCED IN GASES BY RONTGEN RAYS. 



233 



These values are nearly the same as those obtained above for the ions produced by 
Rontgen rays. 

J. S. TowNSEND* has shown that from the ionic velocity in a gas and the coeffi- 
cient of diffusion of the ions in the gas, the value of Ne can be obtained, N being 
the niunber of molecules in a cubic centim. of the gas, and e the charge carried by 
an ion. By comparing this value with that obtained from the electrolysis of liquids, 
the relation between the charges on the ions in the two cases can be determined. 

Using the values of the ionic velocities (v) given in Table XXII., and the corre- 
sponding coefficients of diffiision (K) from the tables given by J. S. Townsekd, the 

^ V 1 ft® 1) 

values of Ne are obtained from the equation Ne = — = for the positive and the 

negative ions in both dry and moist gases. 

The results are given in the following table : — 

Table XXIIL— Values of Ne X lO'i^. 



Gas. 


Moist gas. 


Dry gas. 


Positive 
ions. 


Negative 
ions. 


Positive 
ions. 


Negative 
ions. 


Air 


1-28 
1-34 
1-24 
101 


1-29 

1-27 

M8 

•87 


1-46 
1-63 
1'63 


1-31 
1-36 
1-25 


Oxvfiren 


Hvdrofifen 


Carbonic acid 


•99 


•93 







The corresponding value of Ne obtained for hydrogen from the electrolysis of 
liquids is 1*23 X 10^^ at a pressure of 76 centims. of mercury, and a temperatiu*e 
ofl5°C. 

The values of Ne in the table for the positive and the negative ions in moist air, 
oxygen and hydrogen are perhaps in sufl&cient agreement to justify the statement 
that the charges carried by the positive and negative ions are the same, and that the 
value is also the same for the three gases, and corresponds to the charge carried by 
the hydrogen ion in the electrolysis of liquids. 

The values of Ne for the negative ions in the same three gases when dry are not 
far from those in the moist gases, but the results for the positive ions are consider- 
ably larger. It seems very improbable, however, that the charges carried by the ions 
are different in the moist and dry gases, since most likely the moisture does not in- 
fluence the act of the ionization itself, but either affects the ions after they are formed 
during the production of clusters of molecules around them, or changes the resistance 



VOL. CXCV. — Jl, 



* J. S. TowNSEND, *Phil. Trans.,' A, vol. 193, p. 152. 

2 H 



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234 VELOCITY OP IONS IN GASES. 

to their motion. So if the charges are equal in the moist gases, they should be equal 
in the dry gases also. 

The values of Ne for carbonic acid are all less than that obtained for hydrogen by 
electrolysis, and so indicate a smaller charge on the ions ; but from analogy with 
liquids we should expect that if the charges vary at all, it would be in the ratio of 
one to two or more, unless it is possible to have a charge smaller than that carried by 
hydrogen in electrolysis. 

The writer cannot account for the differences in the values of Ne by supposing 
them due to errors in the ionic velocities obtained, since that would mean the pre- 
sence in the experiments of some error which in some cases influenced the results 
for the positive ions alone, in other cases had an effect upon the values of both of 
the ions, and in still other cases was without effect. 

The experiments described in this paper were performed at the Cavendish Labo- 
ratory, Cambridge, and I desire to express here my thanks to Professor J. J. Thom- 
son for the encouragement and valuable suggestions given in the course of the 
investigation. 



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[ 235 ] 



VI. Undergrmind Temperature at Oxford in tJie Year 1899, as determined by 

Jive Platinum-resistance Thermometers. 

By Arthur A. Eambaut, M,A., D.Sc, Radcliffe Observer. 
Communicated by E. H. Griffiths, F.RS. 

Received May 17,— Read June 21, 1900. 

[Plates 1, 2.] 

Description of the Apparatus and Mode of Reduction of the Observations of 

Earth Temperatures. 

The instruments with which the earth-temperatures given in this paper were 
observed, were five platinum-resistance thermometers of the Callendar and Griffiths 
pattern,* made by the Cambridge Scientific Instrument Company. These were 
purchased by the late Mr. Stone, and were placed in position under his direction 
shortly before his death. 

The method of platinum thermometry seemed to be particularly suitable for this 
class of work, on account of the immunity it enjoys from certain errors attending 
the use of the long-stemmed mercurial or spirit thermometers ordinarily employed 
for underground temperatures. 

A higher degree of accuracy might, therefore, reasonably be expected, and the 
discussion which follows of the first complete year's observations at the Radcliffe 
Observatory shows, I think, that this anticipation has been justified. Some 
discrepancies between theory and observation no doubt appear, but they are of a 
character which seems to indicate a difference between the assumptions on which 
the theory is based and the conditions actually prevailing in the stratum of gravel 
in which the thermometers are buried, rather than thermometric errors affecting 
the observations themselves. 

The thermometers are inserted in undisturbed gravel, the first four lying one 
under the other, in a vertical plane beneath the grass of the south lawn, and within 
a few feet of the Stevenson screen in which the dry and wet bulb, and the maximum 
and minimum, thermometers are suspended. 

In order that the thermometers might lie in practically unbroken ground, the 
following method of placing them was adopted. A pit was dug at the edge of the 

* See the Cambridge Scientific Instrument Ck)mpany's " Descriptive List of Instruments,^' page 20. 

2 H 2 22.11.1900 



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236 DE. A. A. RAMBAUT ON XJNDERGROUND TEMPERATURE AT OXFORD 

grass about 5 feet long by 4 feet wide. One edge of the pit coincided with the 
edge of the grass plot, and the corresponding side of the pit was made as nearly 
vertical as possible. Into this vertical face four iron tubes were driven horizontally, 
the tubes being formed with spikes at their ends to facilitate this operation. The 
tubes are 4 feet long, and into them the thermometers were inserted with the leads 
attached, the mouths of the tubes were sealed up with tow and red lead, and the pit 
filled in. 

The first four thermometers were placed at depths of (approximately) 6 inches, 
1 foot G inches, 3 feet 6 inches, and 6 feet respectively ; but Mr. Stone soon saw the 
advisability of placing another at a lower level, and intended to have gone to a 
depth of 20 feet. But as water was met with at a depth of 10 feet 6 inches, he 
decided to place it just above the water level, at a depth of 10 feet. 

This thermometer was buried, not directly under the four earlier ones, but in a 
separate pit at the other side of the Stevenson screen. This was apparently done 
to avoid disturbing the leads of the thermometers which were already in position, 
but it would have been rather more satisfactory if all had been placed in the same 
vertical plane. 

It is also, perhaps, to be regretted that one or two similar thermometers were not 
buried to considerably greater depths. The presence of water, however, complicated 
matters and introduced conditions different from those which prevailed in the dry 
gravel above. It is not, for example, to be supposed that the thermal conductivity 
or the difiiisivity of permanently water-logged gravel would be the same as that of 
the drier material above it. Hence it would appear necessary to put at least two 
thermometers below the permanent water-level in order to study the flow of heat 
under such circumstances. Besides, it is highly probable that the gravel stratum 
is not very much thicker than. 10 feet. Excavations in the neighbourhood show that 
the blue Oxford clay is likely to be met with at any depth below 12 feet from the 
surface, and in this, of course, the thermal conditions would be likely to prove wholly 
different from those in the gravel. 

The actual depths of the various thermometers as measured in October, 1898 
(when the pits were standing open to enable us to re-standardise the thermometers) 
were as follows : — 

Thermometer .12 3 4 5 

Depth ... 6^ in. 1 ft. 6 in. 3 ft. 6^ in. 5 ft. 8^ in. 9 ft. 1 1| in. 

These thermometei's, with the Callendai* and GriflEiths resistance box, which 
could be connected with each thermometer through a switchboard, had been set up 
as I have stated, shortly before Mr. Stone's death. 

On my appointment to the post of Radcliffe Observer, I took an early opportunity 
of examining the apparatus, and partly with a view of familiarising myself with all 
its details, I proceeded to determine the comparative values of the coils, and to 



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AS DETERMINED BY FIVE PLATINUM-RESISTANCE THERMOMETERS. 237 

re-standardise a spare thermometer which was kept in the observing room for general 
purposes. 

This examination led to the discovery of discrepancies in the readings of the 
apparatus which troubled me for a long time, and which necessitated a large number 
of experiments extending at intervals over the greater part of a year before they 
were traced to their sources and eliminated. 

In this part of the work I have to acknowledge the very generous help and advice 
of Mr. E. H. Griffiths, F.RS., who was kind enough to come to Oxford on more 
than one occasion to place his experience at our disposal, and who, at one stage of 
the investigation, took the resistance box and spare thermometer to Cambridge to 
subject them to a prolonged examination in his own laboratory. 

These discrepancies, though serious in view of the accuracy which we had reason 
to expect from the apparatus, were still small quantities confined within one or two 
tenths of a centigrade degree. They were, for the most part, traced eventually to 
uncertainties in the contacts at the switchboard, and a want of perfect insulation in 
the older leads. These consisted of four india-rubber covered wires which, in the 
underground portion, passed through leaden pipes, but within the observing room 
were without the leaden covering. It was found that these were very susceptible to 
damp, and that the insulation fell away very rapidly when there was much moisture 
in the air, thus giving rise to very puzzling and troublesome discrepancies. 

In September, 1898, the switchboard was improved and new composition cable 
leads substituted, which extended without interruption from the thermometers right 
up to the switchboard. Since these changes were eflfected the discrepancies have 
ceased to appear, except on one occasion (viz., October 27, 1899), when it was found 
that the short flexible lead from the switchboard to the resistance box was 
thoroughly damp. On lighting a fire in the observing room to dry the covering of 
this lead, the irregularities disappeared. 

Since that date up till the end of March of this year (1900) I have kept a gas 
light burning continuously in the room, to prevent the deposition of moisture, and 
have experienced no ftirther trouble of the sort. 

The resistance box is in its general design similar to that described by 
Mr. Griffiths,* but simplified to suit the particular class of work for which it 
was intended. 

It is provided with three principal coils A, B, C, whose nominal values are 
20, 40 and 80 box units respectively, a box unit being about 01 ohm. There are 
two additional coils, one for the calibration of the bridge wire, and another, which 
we have called the " concealed coil," whose value is about 240 box units, which was 
inserted for convenience to balance approximately the resistance of the thermometers 
at 0° C. when the coil A was also in the circuit, so that the reading of the bridge 
wire under these circumstances might be as nearly zero as possible. 

* ' Nataroi* Noyember 14, 1895. 



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BY FIVE PLATINUM-RESISTANCE THERMOMETERS. 239 

oscope is placed on a window ledge to the right of the 
position that the observer can manipulate the commutator 
1 of the current without removing his eye from the eye- 

!it of the apparatus is shown in fig. 1. To the right is 
oroscope ; underneath in front is the commutator, and 
. On the extreme left is the switchboard, and in the 

a small electric motor for stirring the oil in which the 
d. 

the apparatus the method described by Mr. Griffiths in 
1895, was in the main followed. The temperature coeffi- 
\ Griffiths when the apparatus was under examination 
Cambridge. Two separate determinations made in 1898 
e the following results : — 

Range of Temp. Temp. Coeff. 

27 . . 9°-18 0-000242 

} 8 . . 12 -51 0-000240 

bservations, the value 0*00024 has been adopted. The 

been borne out by subsequent observations in several 

ample, the invariable steadiness in the changes of No. 5, 

erature of the box, indicated a high degree of accuracy 

instant. 

he coil values and the unit of the bridge wire scale, the 

lade at the Radcliffe Observatory : — 



)51 


B-A= 19-851 


A = 19-603 


)46 


19-849 


•600 


143 


19-848 


•601 


— 


19-853 


-602 


- 


19-847 





47 19-850 19-601 

Mr. Griffiths* paper referred to above, 

= 80-158-| 

= 39 '979 >mean box units. 

= 19-863 I 

•idge ^wire is equal to 

I '0134 mean box units. 

ible giving the correction for the particular arrange- 



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AS ICmLJUMT 

In iLis eTT«ressJ c K» :s 
the i^ts:st.ii--:>r of :be " vi: 
that or-II in tbe i^:-L-.i 1 
mined fi>:»cn tiie o'teerrh:: 
OMnbined with the cccst.^:. 
sponding resistances R a:. 

Thus, if X be the ralue 
wire when the themior/.r: 
Iwidge wire when it is in.^ 
the total resistances in t:. 
ratio of these resist^LCc^ 

used in the oonstructioa 

03872 
The values of X found 
and 6, 1898, for the purpc* 

Thermci 

1 



4 
5 
A 



For any arrangement of 
the total resistance in the 
correction for temperature 



We thus find the folic 
been used in the observat 



VOL. cxcv. — A, 



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rtVE PLATINUM^RESISTANCE THERMOMETERS. 241 

^efficient, 0*00024, as determined by Mr. Griffiths, 

.^ = R, X 0-00024 X (^ - 14°). 

total resistance in the circuit, and since this includes 
id coil," we require to know approximately the value of 
3 of the equation. This is, perhaps, most easily deter- 
)f the thermometers themselves at 100° C. and 0° C, 
blue found by Mr. Griffiths for the ratio of the corre- 

his coil, Tq that of the other coils in use and the bridge 

s packed in melting ice, and r^ that of the coils and 

3d in steam, reduced to mean box units at 14° C, then 

.vo cases are, X + ^'i and X + ^o> and if we take the 

be 1*3872,* as found by Mr. Griffiths for the wire 

X + r 
this instrument, then rr— — - = 1'3872, and therefore 

^ + ^0 

this way from the observations made on October 4, 5, 
)f standardising the thermometers, are as follows : — 

jr. X. 

240-65 

-65 

'77 

77 

-60 

-65 

Mean 240-68 

lis (Y) and any bridge wire reading (R) we have therefore 
cuit, X + Y + R> and the coefficient of {d — 14°) in the 

(X + Y + R) X 0-00024. 

ng table for the two different arrangements which liave 

s: — 



'Nature,' November 14, 1895, p. 45. 
2 I 



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AS DETERMINED Bl 



Thermometer . 


1. 


1898, October 4 
5 


101-286 


Adopted values , 


101-29 



To those who have standa 
the separate results for the 
however, he pointed out t 
while they were sealed up 
attached ; hence it was imj 
along these tuhes, but for re 
take further precaution agai 
mining the zero points the 
any error arising from this c 

One of the most import? 
degree of permanence in i 
intervals of time. It was tl 
found in my first observe 
previously at the time that 
all the thermometers exhni 
apparatus. This examine ti 
insulation of which was fou 

Another series of discrej 
the switchboard. In the ( 
lead firom the resistance Ik 
also were led the four bn 
By having the steel pron 
finnly pressed against the 1 
was experienced ; and, sine 
the same cause. It has b 
time with the four steel i 
check on the character of t 

Taking advantage of a 
thermometer No. 1 (6 inch 
a year 8 continuous obser 
year, the zero point of thi 
less than 0°005 C, the ac 

I] 
andii 



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FIVE PLATINUM-RESISTANCE THERMOMETEES. 243 
Temperature of Steam. 



2. 


3. 


4. 


6, 


101-474 


101-610 
-589 


101-389 
-367 


101-179 


101-47 


101-60 


101-38 


10118 



Used naked platinum thermometers the discrepancies in 
3. p/s of Nos. 3 and 4 may appear larga It should, 
it it was necessary to standardise these instruments 
n strong brass tubes with heavy leaden-covered leads 
•ssible to eliminate altogether the effects of conduction 
sons given on p. 244 it was not considered necessary to 
st the small errors arising from this cause. In deter- 
thermometers were placed in a trough 3 feet long, and 
,use was very much diminished. 

it considerations in connection with this subject is the 
le fundamental points as determined at considerable 
3 occurrence of discrepancies between the values which I 
dons and those determined about a year and a half 
he instruments were set up, which induced me to have 
3d and to make a thorough re-examination of the whole 
n led eventually to my discarding the original leads, the 
d to fall off very much when they became damp, 
mcies was traced to an uncertainty in the contacts at 
[•iginal form the four steel prongs in which the fourfold 
c terminates, were inserted into mercury cups into which 
58 strips to which the thermometer leads were soldered. 
s amalgamated, and adding springs to keep each prong 
rass strips immersed in the mercury, a great improvement 
I this change was effected, we have had no trouble from 
3n the habit, too, to make the observations from time to 
ongs in both positions, which affords a very satisfactory 
e contacts. 

visit from Mr. Griffiths on October 6, 1899, I had 
s) dug up, and we examined its zero point after exactly 
ations. Determined in the same way as in the previous 
thermometer was found to agree with the earlier value to 
lal values being 

1898 0'806 

1899 0'802 

2 I 2 



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AS DETERMINEE 

A complete determina 
of the sunken thermomet 

(1) The balancing of 

and the temper; 

(2) To R is to be adde 

from Table I. 

(3) The correction to 

Table XL 

(4) The reduction to 

Table III. mult 

(5) The correction froi 

It only remains to re 
the air scale. 

The relation connectin 



in which pt is the platii 
a constant. 

For a completely inde 
the resistance at some 
but the experiments of ' 
value of 8 varies from on 
particular sample of wii 
are given in Mr. Griffit 

The value of 8 for th 
mined at Cambridge t( 
apparatus for the deter 
doubtless have been adv 
this constant. Since, Ik 
variations of earth tern 
within that range the cci 
0*050 (which is quite m 

Writing ^« + dforti 
0'003, its square may be 



T being written £cvptil( 

* 'PhiL Trans./ A, 18S7. 
t Cy. the Report of the C 
Practical Standards for use in 



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' FIVE PLATINUM-EESISTANCE THEEMOMETEES. 245 

of temperature on the platiuum scale by means of one 
is, therefore, reduced to the following simple steps : — 

galvanometer and reading of the bridge wire scale (R) 
L'e {0) of the box. 
le correction for the particular arrangement of coils used, 

luce the bridge wire reading to mean box units, from 

indard temperature (14°). The quantity taken from 
ed by {0 —14) gives this correction, 
^able IV. 

3e the temperature thus expressed from the platinum to 
bese two, established by Professor Callendar,* is 

-'-p"ii^)'-U w 

1 temperature, t the temperature on the air scale, and 8 

ident standardisation it would be necessary to determine 
rd known temperature in order to obtain the value of 8, 
LLENDAR and Griffiths have shown that although the 
ipecimen of platinum to another, it is a constant for any 
References to the original papers bearing on this point 
* article in ' Nature ' cited above. 

)articular wire used in the Oxford instrument was deter- 
)e l'512.t If it were intended to employ the Oxford 
Qation of temperatures over a very wide range, it would 
ble to make an independent determination of the value of 
3ver, the range —15° C. to +25° C. will cover all the 
matures with which alone we are here concerned, and since 
iction does not amount to as much as 0'3, an error of even 
lissible) in the value of 8 would not affect our results, 
equation (6), and remarking that since dj 100 is less than 
jglected, we find 

= 8(t^ - t)/{1 + (1 - 2r)8/l00} 



hiittee of the British Association for improving the Construction of 
metrical Measurements. Bradford, 1900. [September 16, 1900.] 



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X-^ iS 



Ulkf 



^1^ rr^ fc-rc :z. -lijt . 

results. ATii Jis tie : f^r 

disdiict adv^LHt^o? Ir. rv: 
On aocoant of the ir.'i 
carded them alt* -gether, 
pcfftioDS which are alte: : 
vations are taken only oi 
of days in each divisir. 
lengths as small as {k^^ 
thirty and thirty-one c 
intercalating the extra c 



♦ Professor W. Thomsox, 
Eoy. Soc. Edin./ voL 22, p. 
t ' Greenwich Observatior 
t Professor Everett, ' T: 



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LMIMD BY FIVE PLATINUM-RESISTANCE THERMOMETERS. 247 



. . 9-550." 




Sept. 


22 . . 


14-42 C. 


. . 9-59 
. . 9-63 


•CoaA. 




23 . 

24 . 


. 14-41 
14-41 


. . 9-66 






25 . 


14-40 


. . 9-70 






26 . 


. 14-39 


. . 9-75 
. . 9-80 


•CoaR 




27 . 

28 . 


. 14-37 
. 14-35 


, . 9-82 . 






29 . 


. 14-34 



SCoilR 



I 



Coil A. 



Discussion of the Obso^^atiojis. 

the discussion of the observations is to group them into monthly 
) to deduce the harmonic expressions which will represent the 
ermometer throughout the year.* 

be work I have adopted the Fahrenheit scale, as the observations 
educed to this scale for comparison with our other meteorological 
observations of the same kind at Greenwicht and Edinburgh| 
sor Everett are expressed in the same scale, there seemed to be a 
in retaining it, 

e inequality in the lengths of the calendar months I have dis- 
bher, and, as far as possible, have divided the year into twelve 
iltemately thirty and thirty-one days in length. As the obser- 
ily once a day, it is of course necessary to have an integer number 
vrision, but the following scheme makes the differences in their 
I possible, and with one exception, that of January, alternately 
►ne days. In Leap Year this exception would be removed by 
bra day in January, instead of February. 



3N, " On the Reduction of Observations of Underground Temperature," * Trans. 
p. 409. 

^ions/ 1860 (cxciii.). 
Trans. Roy. Soc. Edin.,' vol. 22, p, 429. 



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AS DETERMINE] 

McClellan in the or 
Mr. WiCKHAM or Mr. 
and experience, and, as 
able degree of precision. 
These means were d 
any modification or cor 
to above, except on one 
flexible lead was found 
room. 

This was indicated b 

10 feet thermometer, \ 

steadily, that its readin 

one-twentieth of a degr 

drying the lead, howeve 

the subsequent readings 

as before the diBcrepancj 

As the dampness of 

and in no way affected 

take an interpolated vali 

the difference between 

added as a correction to j 

This particular case i] 

deep sunk thermometer 

apparatua 

The monthly means an 
periods of two months ea 
been selected, as the fir 
the 10 feet thermometer 
in the indications of this 
amplitude of a wave is 
thermometer to the one 1 
In fig. 3 are given 
surface, deduced from th( 
The harmonic expres 
thermometer throughout 



e 



or 0^ao+t 

where t denotes the time 
VOL. cxcv. — A. 



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MINED BY FIVE PLATINUM-RESISTANCE THERMOMETEES. 249 

he ordinary routine, but during his vacation, or on Sundays, 
• Mr. Robinson took his place. All three are observers of skill 
id, as the results seem to show, the observations are of a remark- 
iision. 

ere deduced from the observations as directly obtained without 
)r correction, other than those taken from the tables referred 
1 one date— October 27 — when, as I have mentioned, the short 
mnd to be affected by the dampness of the air in the observing 

led by a sudden change of about 0°*13 R in the reading of the 
er, which, under] ordinary circumstances changes so slowly and 
3ading on any day might be predicted with certainty to within 
degree from the readings of two or three days preceding. On 
^ever, the abnormal readings disappeared by the next day, and 
ings of this thermometer were foimd to lie along the same curve 
•ancy had arisen. 

of the lead disturbed only the reading of the resistance box 

cted the thermometers themselves, we were, therefore, able to 

value for the reading of No. 5 as a standard of comparison, and 

)en this and the actually-observed readings, viz. : 0°*13, was 

to all observations made on that day. 

je illustrates very well the protection which the readings of a 
ter afford against sudden changes occurring unobserved in the 

1 are graphically represented in fig. 2 ; the daily readings for two 
3 each are exhibited in Plates 1 and 2. These two periods have 

first includes the minimum and the second the maximum of 
er, and both illustrate very well the steadiness of the changes 
his instrument, and exhibit also the manner in which both the 

is dinainished, and its phase retarded in passing from one 
e below it. 

n the mean monthly temperature gradients beneath the 
:he same figures. 

ression to represent the temperature of any particular 
it the year will be 

^ = aQ + a^ cos \t 4" % ^^ 2X« + &c. 

+ hi sin \t + 62 sin 2X« + &c (c) 

Pi sin (Xe + El) + P2 sin (2Xt + E^) + &c (d) 

e represented as the fraction of a year, and X is equal to 2ir. 
2 K 



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AS DETEEMIKEI 



i 

o 



08 4a 



u 



o ^5 



»4 



I 



1 ^'^ 



? 



'J 





H 

1 




1 



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•LATINUM-RESISTANCE THERMOMETEES. 



251 




K) X». «J « K •O .«!> 5f' 
•399^ Ul tf9c/9Q 

2k 2 



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DETERMINED 

are here dealin 
nuch importanc 
juracy with ^ 
the formulae i 
between then 

iriBon of Compi 



Thermometer. 


January . . . 


February 






March. 






April . 






May . 






June . 






July . 






August 






September . 




October . 




November . . 


December . . 



; appears that w 
r the formulae, 1 
^ially in the ca^ 
coselves felt to a 
rfece of the gn 
are sunk, beinj 
a fairly uniform 
at will be repres 



K denotes the 
ter below the bi 
ion of this is 



'ing this expres 



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:>LATINUM-^RESISTAITCE THERMOMETEKS. 



253 



observations of a single year, it would be unsafe 
3 deducible from the smaller terms. 
Dbservations are represented by the first three 
by the following table, which contains the 
ily temperatures as computed and those actually 



bserved Mean Monthly Temperatures, C — O. 



2 


3 


4 


■ ■■ 

5 


-0°86 


- d'-55 


- 0°32 


0°-00 


-0-18 


+ 0-07 


+ 015 


+ 0-05 


+ 0-71 


+ 0-48 


+ 0-22 


+ 0-02 


-0-70 


-0-48 


-019 


000 


+ 0-91 


+ 0-25 


+ 0-01 


-0-05 


-1-02 


-0-62 


-0-24 


-002 


+ 1-15 


+ 0-91 


+ 0-49 


+ 006 


-110 


-0-44 


-0-07 


+ 010 


-0-15 


-0-69 


-0-65 


-0-23 


+ 1-87 


+ 1-29 


+ 0-73 


+ 0-07 


-214 


-107 


-0-31 


+ 0-18 . 


+ 1-62 


+ 0-75 


+ 0-18 


-018 



iree deeper thermometers are fairly well repre- 
lonsiderable differences in the two upper ones. 
L, is largely due to the diiUTial variations which 
bout 3 feet. 

le neighbourhood of the spot where the ther- 
lately level and the gravel being, as far as we 
for a considerable distance in all directions, the 
ourier's equation 

0/dx^ = d0/dt . . (c) 

of the gravel, and x denotes the depth of a 



and 2af,finK = — nX. 



Vt = -^' 



le series {d) given oa page 249 we have 
' and E, = /8,a! + y. 



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aiMINE] 
Values c 



Them 
com 


No. 5 a 
„ 6 
„ B 
„ i 
» 4 
„ 3 


Mear 





d the re 
y short 
uried ir 

contaii 
• ^/1^/K 
I those 
dividua 
ote hov 
from tl 
possibly 
h separ 
). 5 bei 
the oth 

"when 
gravel" 
n next 
f close 
y^early 
duced 
)recisic 
ally la 

dedu 
may p 
' whic] 

of th 
it at 



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PLATINUM-RESISTANCE THERMOMETERS. 



255 



iuced from the Half-yearly Wave. 



From diminution 
of amplitude. 


From retardation 
of phase. 


0U45 
•1393 
•1306 
•1287 
•1164 
•1033 


01236 
•1187 
•1129 
•1092 
•1022 
•0949 


0-1271 


01102 


of both , . . 0-1187 



ed from the readings of No. 1, as they seem too 
iations to aflford reliable results. This thermo- 

soil which is of quite a different character from 
ler thermometers. 

rom the annual wave are, of course, much more 
rom the half-yearly wave, and the larger discrep- 
om the latter are not surprising. It is, however 
roborate the others, showing, for instance, larger 
son of Nos. 5 and 4 than from that of any other 
a smaller value of k for the stratum of gravel about 

two thermometers, than for the higher strata, or 
it some distance (9 feet 6 inches) from the vertical 

were open, no very critical examination of the 
nt depths was made ; but it is proposed to repair 
)meters are dug up. 

of the mean values of x/v/k derived from the 
ry remarkable (especially in view of the fact that 
servations of a single year), and seems to indicate 
servations. 

found from the diminution of amplitude, as com- 
he retardation of phase must be traced to some 
ue to the proximity of the Observatory building, 
1 at a distance of 36 feet from the thermometers. 
)eneath the buildings would in all probability be 
epth beneath the exposed surface. There would, 



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5 
4 
3 

2 



values i 
a givei 

to any ^ 
main unc 
oreticalJ^ 
i^e may c 
acted. T 
to 0°-01 1 
)th we h; 
1 therefore 



as the de 



•^1 = 



or the half 
X. = 

at which 
0^-1 F. are 

deals with 
[)aucles l)et 
lave been 
ancles are 
r a periodic 
to he dimi 
physical ai 
ource of in 
lely, thernK 
:he liquid 

—A, 



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PLATINUM-RESISTANCE THEKMOMETERS. 257 

^e, we find 




1 
9 

7 

2 = amplitude of half-yearly wave at surface. 

Df equations (/) we can determine the value of r, 
P,„ or the depth at which the amplitude of the 
, on the hypothesis that the conditions prevailing 
greater depths. 

10 invariable layer so long as equations (/) are 
it an annual variation of 0^'02 F. is less than can 
, therefore, at which the amplitude of the annual 
ntents and purposes be considered as invariable. 
P^ = — 2 (M being the modulus of common 



+ log PoO/M^Wk. 
3h the amplitude of the annual wave is reduced 

rich feet = 66*0 English feet, 

e, 

ich feet = 36-0 English feet. 

al and half-yearly waves are reduced to an 
I similar way to be 45*3 and 21*4 English feet 

tions of a single year, and the results accordingly 
J and observations which, although they are less 
pected, are greater than one would like to see. 
to the fact that the temperature variations are 
as the theory supposes, and as such they might 
he mean of a number of years, and partly to 
I the surface. 

►nsi<lered by Lord Kelvin in his paper, referred 
►rs arising from the uncertainty as to the tem- 
; stems of the thermometers used in Professor 

*> T 



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c^ui. 



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a d 



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Phil. Trans., A, vol. 1.95, PL i. 



It 17 16 19 SO a^ 


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Phil. Trans,, A, vol. 1.95, PL i. 



/d 17 18 i9 20 Sil 22 as B^ £5 25 27 as B9 30 




9 /7 /a /9 20 2/ 22 23 2^ 25 26 27 26 293C 



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VIL The Diffusion of lous ^> 

stance J i 

By John S. Townsend, M 
Fellow of 

Conimunicato 
Rect 

A GENERAL method of findiii 
described in a previous paper,^ 
with ions produced by Rontge 
with ions produced by a radi( 
violet light. Tlie principle of 
from observations on the los 
tubing. 

The experiments were arran 
greater than the loss due to t> 
effects which must be consider< 
binatlon which occurs when tli 
gas ; and the effect due to the 
most of the ions are chargt-'f 
necessary either to correct for 
the experiments so that the lev 

The present paper is divid 
investigation of the relative ii 
and mutual repulsion in causii 
and the results of the experinn 
by ultra-violet light, and by j) 
respectively. The conclusion^ 
Section V. 



In the previous paper we h\ 
distributed throughout a gas, 
* JuHN S. Tow 



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[ 259 ] 



id in Air by the Action of a Radio-active Sub- 
violet Light and Point Discharges. 

hrk'Maxwell Student, Cavendish Lahordiory, 
/ College, Caml/iidge. 

ro/essor J. J. Thomson, F.Ii.S. 

J 17,— Rend June 14, 1900. . 

rate of diffusion of ions into a gas has been 

1 account was there given of the results obtained 

The present paper gives the results obtained 

substance, by point discharges, and by ultra- 
tliod consists in calculating the rate of diffusion 
nductivity of a gas as it passes along metal 

}hat the loss due to diffusion should be much 
uses. In order to ensure this, there are two 
iiig the dimensions of the tubing : the recom- 
x)th positive and negative ions present in the 
repulsion of the ions which takes place when 
electricity of the same sign. It is therefore 
Lirces of error or to arrange the conditions of 
ductivity due to these causes is negligible, 
five sections. The first section contains an 
e of the processes of diffusion, recombination, 
conductivity. The descriptions of apparatus, 
B on ions produced by a radio-active substance, 
larges, are given in Sections II., III., and IV. 
l'a^vn from the experiments are discussed in 

CTION I. 

that when a number of ions. A, are uniformly 
is entering metal tubing, the ratio R, of the 

\. Trans-,' A, vol. 193, 1899, p. 129. 
L 2 8.12.1900 



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BY THE ACTtn 

positive and negative ions ai»| 
experiments on diffusion, we a 
very closely those produced n; 
therefore assume that the ia\\ • 
in the two causes. Tlie method 
explained in the previous pa{M 
the loss due to reconibinatH»n 
sides. 

The time, Zi/V, in the ex\)t 
second, the radius of the tuhi 
finer tubing (a = -5 millini.). 
reduced to ^. The numl)er, 
N to N/9. 

The radio-active sulistance 
the radiation proceeding fron 
smallest that was used in t 
therefore assume that in tht» 
not affect the value of y to th 



When a gas contains ions i 
of ultra-violet light on a nieti 
arising from the electric densi 
It would be difficult to find tl 
of ions in a tube while dift\isi( 
to the error it introduces. 

Let us consider the c<\Ke ol* 
owing to the motion of the io 
we suppose that no diffusion 
electrification at any jx)iiit is 



Pq being the initial density, s 
by unit electrostatic force, 
Po t<> P- 

The proportion of ions lost, 

t John 



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A lUDlO-ACTIVE SUBSTA^^Cfi, ETC. 2()1 

imultaneously in the gas. From the results of the 
to conclude that the ions thus produced resemble 
tgen rays, and carry the same charge. We will 
rning the recombination will not be much different 
nding the correction for recombination has been 
[t was there shown that for small conductivities 
bout 4 per cent, of the loss due to diffusion to the 

;s made with Rontgen rays was about Yb-th of a 

ig 1 5 millims. A new apparatus was made with 

without altering KZ^/a^V, the value of Zj/V is 

ons which recombine is similarly reduced from 

tained in a sealed glass tube, which cut down 
s to produce densities of ionisation less than the 
jriments made with Rontgen rays. We may 

experiments the process of recombination does 

of '5 per cent. 

ual Repulsion. 

n (as in the case of ions produced by the action 
)r by a point discharge), the electrostatic field 
letimes suflBcient to exert a considerable force, 
imount that this effect contributes to the loss 
ng place, but it is easy to find an upper limit 

I gas in a metal tube losing its electrification 
iiies of force from the axis to the surface. If 
)lace, it is easy to showt that the density of 
:he formula — 



Po 



1 -f 4c'Trupf/ ' 

Liforiii, u the velocity of an ion when acted on 
time during which the density falls from 

practically AnpoUt when the loss is small. 
>, ' Phil. Mag.,* Juiie, 1898. 



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Dounda 

r down 

d by tl 

a^ to tl 

th air at 

he mova 

be of tl 

3 tightly 

leading t( 

atus. W 

3 tubes w( 

•elocity of 

gasometei 

3 suljstanc 

r), and th< 

of thin gl 

f wire 8up] 

it by the ac 

through th( 

C was seale 

i radio-activ< 

ebonite supj 

haken when 

was eonnecte 

eing to earth. 

the other pa 

rth. The rod 

metal screens 

le electrometer 

abe A is in n 

ipt E, there is ii 

es T into the s]); 

dctrode, and a d 

ordinary conditi* 

turbulent moti< 

itial difference ( 

periment never < 

ionisation and \ 

larging A to 40 

one sign are en 

:o the number of 



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F A RADIO-ACTIVE SUBSTANCE, ETC. 263 

the disc. When either set of tubes was pushed 
arge tube divides itself equally among the twenty- 
bes T. The disc a^ was soldered to the front of the 
nt of the tube B^j. 

ospheric pressure the stream of air was obtained by 
ylinder of a gasometer. For experiments with dry 
asometer was connected to wide tubes of calcium 
ked with glass-wool was put between the drying 
so that particles of dust should not be carried into 
it was desired to make experiments with moist air, 
emoved, and long tubes half filled with water were 

air along the tubes T could be varied by changing 

as obtained from E. de Haen (Chemische Fabrik, 
eparation labelled " Radio-active Substance A " was 
containing some of the radio-active substance, was 
s inside the tube A as shown in the figure. The 
substance was much more intense than uranium rays, 
iss tube was strong enough to ionize the surrounding 
1 order to prevent any moisture from coming into 
ibstance, which was deliquescent. The tube A was 
5, S, to the top of a heavy box, so that the tube C 
tubes B| and IB^ ^^® fixed in position, 
o one terminal of a battery of forty lead cells, the 
i'he rod F was connected to one pair of quadrants of 
f quadrants and the case of the electrometer being 
and the wire connecting it to the electrometer were 
\ that external electric charges should not give any 
tie. 

illic connection with all the parts of the diflRision 
electric force acting on a stream of gas until it comes 
between E and B. The air takes about one second 
3rence of potential of a few volts between E and B 
J, suffice to collect all the ions of one sign on E, but 
of the gas as it escapes from the tubes T, a much 
) volts) was used. The potential of the electrode 
needed 1 or 2 volts. It was found under similar 
ocity of air that the electrometer deflection was not 
^Its instead of 80. We therefore conclude that all 
^cted on E, so that the electrometer deflection is 
)ns that come through the tubes T, 



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BY THE ACTiC 



nds of the tubes 1 
tube A, and was 
A CQuld be found, 
apillary tubes K, 
le admission of ai 
ary tubing was coi 
made with wide t 
vessels W^ and W. 






er to obtain a streai 
between suitable lin] 
iusted until the pn 
^as connected to tht 
IS adjusted so as to 
a open to the air. 1 
as turned on for a 
3re desirable necessil 

out by the water-pu 
f rose at the rate of a 

the pressure was as 
jnt The velocity V 
>f air that escaped fr 
5X0V,— A. 



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N OP A feAbiO-ACtlVE SlJiBSTANCE, ETC. 



265 



I and Bjj. A short brass tube L was soldered near the 
onnected to the manometer M, so that the pressure of 

The air from the room was admitted to the apparatus 
laving first passed through a tube of glaiss-wool G, to 
Y dust which might alter the resistance of the tubing, 
lected to the drying tubes, and the rest of the connec- 
:bing. The tube u leading from B was connected to 

which were exhausted by means of a water-pump. 



^£ 









^i>^ 



1 



~ g y 



Fig E. 



of air through A at a given pressure P, and with a 
s, the stopcock Si was closed and the whole apparatus 
mre was a few millims. below the pressure P. The 
elivery tube of a gasometer, the movable cylinder of 
3 on the point of moving downwards when the gaso- 
j vessel Wg was then connected with the water-pump, 
V minutes. The velocities V (through the tubing T) 
ed a larger supply through the apparatus than could 
) so that the pressure, as shown by the manometer, 
ut 3 milliius. per minute. The stopcock Sj was turned 
ich above P as it was below P at the beginning of the 
the tubes T can be accurately found by observing the 

the gasometer and the time during which the stop- 
2 M 



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" THE ACT 

roa of men 
a the tempe 

jitive Ions ii 


V- 1 J 


344 Vt 


387 55 


420 4C 


410 30 


682 20 


itive Ions in 


V. P. 

1 


368 


77: 


430 


40( 


609 20( 


n bet^ 
icaUy I 


^een t 
»y me? 


1 




^ 


X 




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N OF A RADIO-ACTIVE SUBSTANCE, ETC. 



267 



ry ; V is the mean velocity in the tubing Tj in centime, 
ture of the air during the experiment : — 



Dry Air. 



Table II. — Negative Ions in Dry Air. 



a 




»i. 


na. 


V. 


P. 


e. 


19 


63-4 


138-6 


344 


772 


19 


13 


43-0 


93-8 


387 


650 


13 


16 


24-8 


680 


420 


400 


16 


13 


10-6 


39-9 


410 


300 


13 


1 12 


7-6 


31-5 582 


200 


12 



[cist Air. Table IV. — Negative Ions in Moist Air. 



a 



18 



11 



9-5 



«1. 


«2. 


V. 


P. 


e. 


711 


135 


368 


772 


18 


21-0 


66-3 


430 


440 


11 


7-6 


27-1 


609 


200 


9-6 



I ratio y {= ^'i/wg) and the coefficient of diffusion can 
3 of a curve representing equation 2, Section I. The 




9 i'O hi i'B i'3 h^ h^ 



3 M 3 



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BY THE ACT] 

•es, Si and S^ were 
A quartz-plate, 
e the joint air-tij 
Jaced in the wii 







^ 



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ON OF A RADIO-ACITVE SUBSTANCE, ETC. 



269 



saddled on to it, each of them surrounding one of the 
J, was fixed to the end of S^ by means of sealing wax, 
ht, and a piece of wire gauze, having the same curvature 
dow Wi, completely filling it. A piece of zinc, Z, of the 




3ce of brass which was cut out of the window Wg, was fixed 
sed through the ebonite disc D. The disc fitted tightly 
joint was made air-tight. The zinc did not touch the 
il relative to A could be varied as desired. When ultra- 
he quartz and the gauze, it falls on the zinc, and negative 
face of the metal. Some of these ions can be sent into a 
^ A by lowering the potential of the zinc relative to A. 
battery, H, was insulated and its positive terminal was 
.tive terminal to R. 

minium wires was used as the source of ultra-violet light, 
g the spark was contained inside a box covered with lead, 
through which the light from the spark fell on the quartz- 
nals of the secondary of a Ruhmkorfi* coil was connected 
joyden jar, and the other terminal to the inner coating, 
e coil, and the discharge took place across the spark-gap 
ium wires. The air in the neighbourhood of the spark 
id, so that it was found necessary to pack wool round the 
rifled air from coming into the neighbourhood of the rod F 
: to the electrometer. When this precaution was taken it 



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BY THE ACT 

The density of ionizatioi 
has to be polished from ti 
smalL In the above exp 
the third experiment with 
gas in the tubes T^ is 5^7 
apparatus per minute was 

On standardising the el 
sponded to a charge of Oi 
fication p was therefore { 
Section L that the product 
due to self-repulsion shouL 
the sides. In the present 
need be made for the loss c 

Secttion I\ 

In order to make the a 
charge, the changes shown 
diameter) were made in th< 



F%g.4 



^h 



'h 



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OP A RADIO-ACTIVE SUBSTANCE, ETC. 



271 



pends greatly on the state of the zinc surface, which 
bo time in order that the ionization should not be too 
ents the greatest density of electrification occurs in 
ga-s. The mean time, ^, spent by any portion of the 
ad. The total volume of gas that passed through the 
> c.c. 

ometer, it was found that each scale division corre- 
electrostatic unit. The mean density of the electri- 
< 10"^ electrostatic unit per C.C. We have shown in 
C t must be less than 10"* in order that the loss of ions 
B less than 1 per cent, of the loss due to diffusion to 
e the product pX* is '9 X 10"^\ so that no correction 
to self-repulsion. 

-Ions produced by the Point Discharge. 

stratus suitable for experimenting with the point dis- 
L fig. 4 were made. Two circular holes (1*6 centims. in 
ibe A, and two tubes, Q and B, of the same diameter as 




;,£^- 



Fig. 6. 



w 



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\r * - • 

-S.^:^-- as 1 



J>anie as 
iitinis, fn»i 

xjx^riineiit 
burth ex J I 
:es place i 



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S OF A tVPKV.ACriYK srRSTANCR KTX\ l^7S 

I stcvl \\i\\t in the tiiK* Q, the jxMiit Knr*g at :V»e 

e ui:der5«ine ivndit unis as exj>oriment K exivpt tV,ai 
• :\iVr that the ::[5u^ v^houM have a siiialler eUvtnt;vxu:ov, 

vriinents 2 and 3, with a platimim jvnnt sul^titiuoil 

.< held in the tul^ A hi the [xv^tiou shown ux ti^j. o, 
ermient 1, except that the }x>int n>-5U5 dniwu up tho 
111 the aperture iu A* 

»iily ones iu which the efteot itf st*lf-n*pxilslon may 

;s of ions in the tube Tj, so that the n^Uuos of K 

lav be a little too big. 

*ent. which occurs lietweeu the vahuvs obtainiHt in 

»s obtained iu experiments 2, 3, and 4, is pix>lvibly 

ifference in the ions, 

iiu of air in A from a |x>iut some distance up tho 

5, as Experiment 6 sho\>T5 that they dittuse nunv 



-Xegati 


ive Ions 


in Dry 


Air. 


«i. 


fi^. 


V. 


K. 


■2 


138 


337 


•0382 


62 


165 


326 


•0367 


<-5 I 


150 1 


323 ' -0368 


■2 i 

1 
I 


160 


342 


•0324 



[)oiiit ill the tul)e Q, tho point Wnug at tho 

1, with a platinum point Hul>Hlitutod for [\\v 

I the tube A, as shown in fi^. 5. 

7, exct»pt that the j)oint was drawn uj) \\\\' 
ii-e ill A. 

cjilly the Hamo vahioH for tho ooollicioni of 
vH that larger ions aio imxhuvd when llio 
tube Q. 



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BY THE AC 



The theory of the inter 
iversely proportional to tl 
een confirmed by the exjv 
Section IL show that the 1 
onsists of ions. The prt 
ontribute to the total pn 
ressnre in this case is the 
:e see that between the 
iffiision is inversely propo 

We conclude firom this t 
aries between these linlit^ 

L 

' The experiments on diti'i 
ctive substances, and ultn 
he same changes arising 
oefficients of diffiision of i 
ases which are greater thu 
The ions produced by th 
ther methods, since theii 
egative ions in moist air. 

Coefficients of Dit- 



Method. 



Rontgen rays 



Radio-active siil>stance 



Ultra-violet light 



Point discharge 



Me^ 



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OF A RADIO-ACTIVE SUBSTANCE, ETC. 



275 



IX v.. 



of Pressure. 



an of gases shows that the coefficient of diffusion is 
1 pressure of the two diffusing gases. This law has 
ts of LoscHMiDT and others.* The results given in 
1 be extended to the case where one of the gases 
that the ions exert is so small that it does not 
\y an amount which could be measured. The total 
ire 0^ the gas into which the ions are diffusing, and 
as 772 and 200 (millims. of mercury) the rate of 
;o the pressure, 
size of an ion does not change when the pressure 



iced hy Various Methods. 

w that the ions produced by Rontgen rays, radio- 
ight are nearly of the same size, and subject to 
) presence of moisture. The following table of 
lir shows that there are differences in the various 
light arise from experimental errors, 
ischarge are larger than those produced by the 
diffusion is much slower, except in the case of 



ons produced in Air by different Methods. 



Dry air. 


Moist Air. 


) ions. 


Negative ions. 


Positive ions. 


Negative ions. 
•035 




•043 


•032 




•043 


•036 


•041 




•043 


— 


•037 


1 


•037 
•032 


•028 
•027 


•039 
•087 



Theory of Gases,' Chap. VIII. 
? N 2 



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[E AC 

y that 

It of tl 
> come 
iits of 
^ woul< 
ficient? 



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OF A RADIO-ACTIVE SUBSTANCE, ETC. 277 

sizes of the ions produced by point discharges vary 
iparattis in the neighbourhood of the point. What is 

definite conclusion with regard to the charges is to 
sion and the velocities of ions produced under similar 
possible with the apparatus I have used for the deter- 
[iflEusion, and I hope to be able to make observations 

to an accurate determination of Ne. 

charge on an ion in a gas in terms of the charge on 

is of some importance, as it enables us to obtain 
f electricity. 
IS a similarity between the minimum subdivisions of 

gases. 

r the determination of the charge in absolute units 
b gases, and since all the determinations depend upon 
^reat accuracy cannot be expected. The results show 

is of the same order for ions obtained by various 
)een obtained by Professor J. J. Thomson for ions 
Lnd by ultra-violet light ;t the values are nearly the 
^ and 7 X 10"^^ electrostatic unit. These values do 
due 5 X 10~^^ which I obtained for the charge on the 
en off by electrolysis.;}; 
;e is the same in all cases, we must assume that the 

in order to account for the differences observed in the 
LELLAND,§ by examining the velocities of the ions 
owing wires, found that the mass attached to the ion 

circumstances connected with the ionization. The 
es for small differences of temperature of the wire, 
llects round an ion is very variable. We would not 
dio-active substances would have an effect upon the 
icy to collect round a charged ion, but it is possible 
Afferent ways by different kinds of rays, so that the 
h point discharges in air there are actions taking 
e the carrier increase in size. Thus the oxides of 
t condense round the charge and lower the rate of 
of the ion. 
-violet light has any effect on dry air, but Wilson|| 

Phil. Mag.,' Dec, 1898. 
Phil. Mag.,' Dec, 1899. 

* Phil. Mag.,' Feb., 1898. 

1), * Camb. Phil. Soc. Proc,' vol. 10, Part VL 

* Phil. Trans.,' A, vol 192, 1899. 



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279 ] 



%re of Metah, (Second Paper.) 

Mechanism and Applied Mechanics in the 
TER RosENHAiN, B.A., St. John's Colhge, 
h Scholar, University of Melbourne. 

ead, in Abstract, May 31, 1900. 

:s 3—13.] 

3resent paper deal principally with the 
>ntinuation of the research described in the 
' A, voL 193, 1900, pp. 353-377). In iron, 
•een studied with the aid of the microscope 
D, Chakpy, Stead, and Roberts-Austen 
esult of their labours it is well known that 
nent of the crystalline grains of the metal. 
1 in tension its crystalline grains become 
; when the specimen has been subsequently 
all signs of such elongation disappear from 
microscope. * In fact it is not generally 
Btween the crystalline pattern seen in the 
ling. In general, the pattern seen after 
ilar specimen before it has been strained, 
em- produced depend very much on the 
trly upon the temperature applied, the time 
g. Arnold and Stead have shown that 
rge crystals in iron and steel. But even 
e is well known to produce complete re- 
that these changes occur at critical points 
the cooling of the metal. These arrest- 
is natural to suppose that they are evi- 
of the metal. 

! hoped to observe this change taking place 
jrimental difficulties of keeping a specimen 
J being heated were successfully overcome, 
of iron failed. 

10.12.1900. 



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^E STEUCTUEE OF METALS. 281 

r the atmosphere. On leaving the apparatus 

ugli the window by means of " vertical " illu- 
ve itself; as we were content with moderate 
n objective of long focus could be used. 
, we did not succeed in keeping the polished 
» had been reached ; but in the course of our 
)n was observed. On beginning an observation, 

'' ferrite " grains could be clearly distinguished. 
t'^ere then slowly raised by gradually turning on 
irst visible change was a dimming of the image, 
etely blotted out. This we supposed to be due 
3 part of the optical system, but we could not 
re further, the image of the crystals reappeared 
ting off the reflected light, the metal could be 
mating still further, the pattern was rapidly and 
5 of the surface ; the metal was now dull-red. 
ily dark spots appeared, and spread rapidly over 
eed at which they spread could, however, be 
leating current. The spots appeared well in the 
)parent darkening could only be pushed to the 
erably higher temperature. On allowing the 
isible, either on passing through this range of 
i ; nor could the phenomenon be made to recur 
d below redness; but, if this was done, the 
y in the same specimen. It seems probable that 
ance occurs in the metal itself and not merely in 
Lis in this film remain entirely unaffected by it, 
'ssion that he is looking at an action taking place 
arent film. On repeating these observations with 
atmospheres than hydrogen, no such phenomenon 
to suppose that the phenomenon is a result of 
'^drogen and the iron. From its occurrence just 
sspond to the arrest-point, about 487^ C, discovered 
^s Research Report,' Inst. Mechan. Engin., 1899). 

hydrogen caused the surface of our specimens to 
rercome this difficulty by observing the surface of a 
ating was again done electrically, either by passing 
ecimen, or else by placing the specimen in the centre 
and on a piece of terra-cotta. In both cases the 
, the electrodes passing through a sealed cork at the 
2 o 



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^E STRUCTURE OF METALS. 283 

cloned the attempt to observe this process iu 

to the study of similar processes of amiealing 

iible metals, particularly lead. 

-ture required to produced re-crystallisation in 

we observed in specimens of plumbers' sheet- 

ite nitric acid. When thus treated, ordinary 

Laiit crystalline structure on such a large scale 

The etched surface shows all the appearances 

light on etched crystalline surfaces ; when the 

he various crystalline grains in turn, the colour 

each grain, but different on different grains. 

ich a surface magnified two diameters : these 

ring, and must therefore be observed and pho- 

►lution. 

•face reveals a peculiarity in the configuration of 

jn to have many remarkably straight boundaries 

of parallel boundaries being frequently observed. 

ible what we had previously observed in wrought 

[uent occurrence of twin crystals. In our earlier 

presence had always been readily detected by the 

in them by slightly straining the specimen after 

observed in sheet lead by this method has been 

Phil. Trans.,' A, vol. 193, 1900, Plate 26, fig. 40). 

of detecting twins is not available, as the rough- 

iepth of etching employed make it impossible to 

mce of twin lamellsa nevertheless becomes evident 

with oblique light. Fig. 3, Plate 3, is a photo- 

B magnified 40 diameters. The figure illustrates 

imination, which has picked out a few isolated 

J while neighbouring ones remain almost dark. 

luminated grains, a number of dark patches are 

- boundaries occurring in parallel sets which are 

is instance there are three distinct parallel sets 

hey are twin lamellae becomes apparent when the 

.ted, thus altering the incidence of the light. As 

Qs that were bright become dark, but presently 

eviously dark shine out brilliantly, all the bands 

ashing out simultaneously. Fig. 4 is a photograph 

rotation of about 30°, and illustrateis this appear- 

'^hich catch the light simultaneously are evidently 

which the orientation of the elements has been 

other words, they are twin lamellsB. 

2 o 2 



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CRYSTALLINE STRUCTURE OF METALS. 285 

ermine the effect of very severe strain on the crystalline 
a soft, ductile metal, plastic deformation may be carried to 
le adaptability of the individual crystals to change their 
leavage planes may be insuflficient. Careful observation of 
f a piece 'of lead under severe compression confirms this 
it the crystals are gradually flattened out in proportion to 
specimen, but when the " flow " becomes considerable it is 
ady very thin and flat, are driven into and through one 
ig in a grain or structure which is small, but still entirely 
inalogous to what occurs in the fracture of a more brittle 
;hat in a more brittle metal, when " slip " has gone so far 
rystal, the new siu'faces thereby brought into contact do 
cture results ; in lead the freshly exposed surfaces do weld 
re, a fact which is associated with the possession, on the 
utility. Fig. 6 is a micro-photograph showing the crystal- 
i lead magnified 1 2 diameters, while fig. 7 shows the much 
3shly and severely-strained lead magnified 30 diameters, 
xperiments with lead, the process of straining was carried 
of the metal in a compression-testing machine, letting 
jlock, originally about 1 inch high and f inch diameter 
ut I inch thick. 

e changes in the crystalline structure of such strained 
of taking a series of photographs of a marked area of 
;ime during which the metal was exposed either to the 
room or vras subjected to special thermal treatment. 
iken the surface was thoroughly re-etched ; our experi- 
L had convinced us of the necessity of this proceeding, 
pecimens have confirmed the previous experience. In 
t in any way produce a visible change in the surface 
ad been resorted to, and fairly deep etching is required 
tirely. This applies more particularly to the channels 
ater-crystalline boundaries — ^these may often be seen 
y-formed pattern, but quite independent of the new 

ted of alternate applications of concentrated and very 
, where very deep etching was required, an electrolytic 
great advantages of dealing with a metal like lead 
jrystaA& ; by enabling us to use deep etching it allows 
sed vsrith, and it becomes possible to obtain micro- 
ns, and under oblique light, which exhibit clearly the 



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LINE STRUCTURE OF METALS. 287 

erefore be taken as no more than an extremely 
agnitude with which we are concerned in these 
iperatures. 

sheet-lead showing fairly large crystals as an 
perature annealing has continued for a long time, 
plying a higher temperature, so as to determine 
occur. Our observations show that the metal in 
•ately high temperatures, three minutes' exposure 
ture of 200° C. being suflficient to produce a great 
If the specimen be kept at 200° C. for a long time 
it becomes very slow, and ultimately a state is 
»erceptible. 

inclusive are a series of micro-photographs of a 

1 4 shows the appearance of a typical specimen of 
iange produced by 30 minutes at 200° C. Except 

is very diflficult to trace any connection between 
ginal. Fig. 16 shows the same surface, re-etched 
)^ C There has been further change, but not to 
n the first half-hour. The change is most marked 
side of the figure ; in fig. 1 5 it shows a mottled or 
nes filled in in fig. 16, while there is a considerable 
ss of the two tongue-like projections that start on 
nd. 

crystal is seen in fig. 17, which was taken after 
^ C. Here another twin band has become evident, 
htening of the boundaries has taken place. This 
n fig. 18, taken after four days' further annealing. 

representing the final state of this specimen, as 
1 no further considerable change. This specimen, 
ng feature, which we have often observed in other 
)r photography occupied the centre of the surface 
1 approximately f inch square by ^ inch thick, 
marked area did not show by any means the best 
1. In this case, as in many others, we found that 
ing crystals were formed at or near the edges of 
it the same stage as fig. 18, and with the same 
irked area, shows the remarkable development of 
men. 

uch a series of photographs, one consideration must 
3nce produced in the appearance of the surface by 
f incidence of the light. In spite of the utmost 



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CRYSTALLM STRUCTURE OF METALS. 289 

men; the following photographs were taken after the 
to 200^ C. for the time shown in the table : — 



amber. 


Days. 


Hours. 


21. . . 


— 


17-5 


12. . . 


I 


17-5 


3. . . 


2 


17-5 


4 . . . 


5 


16 


5 . . . 


39 


20 


1 . . . 


39 


20 



icture characteristic of freshly-strained lead, with one 
ave persisted from the original crystallisation. In 
rrown considerably, and a general change of pattern 
g feature is the large skeleton crystal that hafi 
/ corner of the marked area. This skeleton is seen 
1 figs. 22 and 23. Figs. 24 and 25 were taken under 

in order to show another large crystal which gave 
ig. 24 it is still somewhat skeletal, but in fig. 25, 
3onsolidated, all its outlying arms have disappeared, 
lefined crystal, part of which is seen as a dark arm 
h of the specimen at the same stage as fig. 25, but 
jrs), and so illuminated as to bring the new crystal, 
, into brightness. This new crystal is seen to be 
irs, and from its position relatively to the marked 
ime crystal whose early stages are seen in figs. 22 
3lJent example of what may be called an aggressive 
3, also at 8 diameters, is shown in fig. 27, Plate 10. 

men can be seen, and the photographs illustrate 
stals are generally near the edges of the speci- 
3 Jarg-e crystals are not mere surface layers, but 
ess of the specimen, and can be readily identified 
s case, the specimen is a plate about one-eighth 

J in the annealed metal is apparently in no way 

3,1s in the original state before straining ; the 

in a specimen whose original crystals were 

by the photographs (figs. 26 and 27) of these 
ce of twin crystals, both as inclusions in the 
3S. In fig. 26 three distinct sets of straight 
2 i> 



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rSTALLINE STKUCTURE OF METALS. 291 

th oi ciystals occur in lead only when the metal has 
)vere plastic strain. The structure of a cast specimen 
ures which cause a strained specimen to show rapid 
i arranged to cause rapid cooling, specimens of lead 
lute crystalline structiu'e, whose scale is not very 
Tushed lead ; such a specimen was exposed to 200° C. 
sibJe change of structure occurred. A piece of this 
severe crushing, and on further exposure to 200° C 

specimen is cooled from temperatures of 200° C. to 
r-temperature has no visible effect on the structure. 
has no visible effect ; quenching in water, cooling in 
were all tried on a number of specimens without 

3ned to a small extent by severe strain, and the 
^storing softness is correspondingly small. In one 
^ad was crushed under a given load in the testing- 
mtil no further creeping occurred. The specimen 
i under the same load, when a distinct amount of 
lace. 

)ed above as having been made with lead were 
nt lend themselves to similar treatment ; those 

well shown when a surface of a cast ingot of the 
iJoric acid. These crystals are generally large, 
s obtained on etching the surface of commercial 
nter-crystalline boundaries may be seen on the 
e grooves or channels. The presence of these 
16 method of manufacture, during which these 
ted tin, and allowed to drain. Ab the plate is 
it crystallises, but any fusible impurity present 
' longer, and, being forced by the crystallising 
s, the still fluid impurities will drain off, thus 

i of comnciercial tin-plate is shown in fig. 28, 

lalf the natural size. In this photograph the 

irly seen, but it also illustrates another and 

ixll ca^es of an etched crystalline metal viewed 

v^ed that, under a given incidence of light, 

^were dark, and that the illumination was 

;tal. In the etched tin-plate this is not the 

l» 2 



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TALLINE STRUCTURE OF METALS. 293 

f of true crystalline plates ; such distortion would 

in the coefficients of expansion of iron and tin 

rate of cooling. Considering the extreme thinness 

t of distortion might well be purely elastic and 

crystals of tin. 

specimen, but to the same scale as fig. 31, illustrates 
and arrangements of the tin crystals that can be 
^ solidification ; the crystals in fig. 31 were formed 
>se in fig. 32 by quenching the specimen in water 
^as still melted. By means of local quenching and 
attems can be obtained ; such processes have long 
nufacture of what is called *' moirSe metallique." 
it the small crystals of tin which are obtained by 
water do not show any growth when the metal is 
?mperatures short of the melting-point. Even a 
does not make them grow or re-arrange themselves, 
a', be reduced to a minutely crystalline structure by 
lens so treated we have observed re-crystallisation to 

ts on the re-crystallisation of cadmium at moderate 
can be strainec^ by compressiou until its crystalline 
gh interpenetration of the original larger crystals. 

(12 diameters) photograph of an etched and marked 
strained piece of cadmium. Fig. 34 shows the same 
exposure to 200® C. It now shows a well-defined 
hows the same area again, after six days' further 

considerable increase in the size of the crystals is 
;he gradual growth of some of the crystals is very 
itures that we have observed in the case of lead are 
1 we can see no invading branches and no aggressive 
;o be any considerable amount of twinning, 
just described were also made on specimens of zinc, 
* zinc strained by compression at ordinary tempera- 

on exposure to 200*^ C. Some results obtained with 
electric batteries, were particularly interesting. It 
lechanical properties of zinc are widely different at 
ly that the metal is soft and ductile at temperatures 
it is generally worked at that temperature, while it 
at and above 200'' C. Commercial sheet-zinc, rolled 
emains fairly soft and flexible at ordinary tempera- 
e is too minute to be seen in specimens etched with- 



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TALLINE STRUCTURE OF METALS. 295 

has been very widely believed that annealing or 
iron and steel, are "critical" phenomena which can 
^finite temperatures. Arnold has gone so far as to 
such an " annealing point." Various of the " arrest- 
and steel have also been regarded as representing 
Baling, but the connection between the two is by no 
uud phenomena of annealing or re-crystallisation in 
e interesting ^o inquire whether any corresponding 
the cooling of these metals. We investigated the 
trie arrangement consisting of two thermo-electric 
ionval galvanometer, and a potentiometer somewhat 
OBERTS- Austen ; the deflections of the galvanometer 
us of a telescope and scale, instead of being photo- 
)e, therefore, that either from this cause, or from 
whole arrangement, some minute arrest-points were 
Iting-points and the ordinary temperature of the air 
observed in the three metals tried, i.e., lead, tin and 
f they exist at all, may be found at much lower 
1 our experiments were carried, 
te the phenomena of re-crystallisation in lead, &c., 
1 heat is evolved during the cooling of the metal, 
xt even in iron the arrest-points are not necessarily 
ing, we look for a theoretical explanation of these 
he theory of re-crystallisation which we shall now 
s for the explanation of the phenomena described 
lit part in the action to the impurities present even 
ies which we believe to be of importance are those 
utectic alloys, or fusible compounds, with the metal 
nainly metals, particularly the more fusible metals, 
, mercury, sodium, or even rarer metals, such as 
when a metal containing a small proportion of such 
[ties are, for the most part, segregated in the inter- 
•^stals themselves form at a temperature when the 
lid, and the growing crystals gradually push the 
laries. Where the quantity of impurities present is 
n be seen under the microscope forming an inter- 
where the " pearlite " plays the part of a eutectic, 
^ucture ; other examples can be found in the gold- 
^ Messrs. Heyoock and Neville.* Where the 
very small, the meshes of inter-crystalline cement 
YCOCK and Neville, *Phil. Trans.,' A, vol. 194, plates 4, 5. 



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^ALIJNE STPvUCTURE OF METALS. 297 

\9 over different crystals, in such a way as to produce 
een adjacent crystals. Another phenomenon, seen 
ic acid, is also of interest in this connection ; it has 
esent paper (see p. 284). We there have a case of 
and deposited upon another crystal in its proper 

ik it must be admitted that different crystal faces, 
heir elements, differ in solubility in the same solvent. 

difference is a further step in speculation which is, 
connection. Such differential actions may, however, 

differences of electrical potential in the surfaces 
w of the matter, then the diffusion across films of 
olysis. Now, while diffusion in metals and alloys is 
ectrolysis in an alloy has not yet been demonstrated 
land, the close analogy with salt solutions leads one 
ctrolysed, and those who have experimented in the 
rtain that greater experimental resources will not 

5 phenomena of re-crystallisation which the solution 
oes not cover, while the electrolytic theory explains 

> fact that only strained crystals will grow, while 
ency to change even at higher temperatures. The 
eory, is that in the unstrained state the crystals are 
uous films of eutectic, and that electrolysis only 
itortion has broken through these films in places, 
ne into contact ; the electrolytic circuit would then 
:>ne crystal to the other by direct contact and back 

*e-crystallisation in solid metals may be summed up 
I is one of solution and diffusion of the pure metal 

> fusible and mobile eutectic forming the inter- 
results in the growth of one crystal at the expense 

n solubility of the crystal faces on opposite sides of 
bable that this phenomenon of directed diffusion is 



:iKR (*Comptes Eendus de TAcad^mie des Sciences,' vol. 116, 
Dii into iron is affected by the action of an electric current. He 
► iron electrodes enclosed in a fire-clay tube ; the whole was 
iperes was passed for three hours, when the anode was found 
idorgone considerable cementation. This action in the interior 
carbon-iron eutectic. 
2 Q 



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ALLINE STRUCTURE OF METALS. 299 

of the weld and there ended quite abruptly. It 

that this weld line was mechanically weak; it 
icult to cut or tear the metal along the weld as in 
erefore behaved as a true inter-crystalline boundary, 
tectic, and therefore forming a barrier to crystalline 
earance of such a weld in section after annealing 

30 diameters. The line AB is the weld. As these 
Qiercial lead, we were prepared to find that, as a 

eutectic would have occasionally found its way into 

seems to have happened only very rarely. We 
nd only in two instances did we see a slight amount 
1 line of the weld. We think that we are justified 
IS to the accidental presence of impurity. 
If we have in a welding surface an inter-crystalline 

crystalline growth owing to the absence of eutectic, 
)lied, growth should occur there as elsewhere, 
erpose a thin but continuous layer of lead-bismuth 

in welding ; the specimen was then annealed for 
above the melting-point of the eutectic — but on 

the layer of eutectic had persisted as such, and 
lit in this case the film of eutectic introduced at the 
litions were therefore analogous to those which hold 
ystals, where, as we have pointed out, growth does 
nent conclusive it was necessary to have a discon- 
weld. We accordingly tried another experiment, 
3S of the same alloy, and after aimealing we found 
►ssed the line of the weld in many places. This 
times, various impurities being used, such as the 
c, pure tin, cadmium, bismuth, and mercury. All 
)wing considerable growth across the weld after 
t the amount of growth observed varied very much. 
3 of crystals that have grown across the weld ; the 
iicated by a discontinuous line, CD, probably repre- 
i non-metallic character, around which the crystals 
ound the slag in wrought-iron. 
I to the action of the impurities which were intro- 
ite the possible contention that their action was 
of the nature of that of the " dirt " more or less 
ns, certain further experiments on welds in lead 

introduced at the weld was — 
2 Q 2 



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TALLINE STRUCTURE OF METALS. 301 

of a single specimen of freshly crushed cadmium, 
continued exposure to 200° C, under oblique light, 
• 12 diameters. (Plate 12.) 

sheet-zinc by exposure to 200^^ C, oblique light, 
ers. (Plate 13.) 

I weld in lead, using clean surfaces, after prolonged 
r the weld is seen at AB. Oblique light, magnifica- 
late 13.) 

I weld, using eutectic in the weld, after prolonged 
>f the weld is at CD. Oblique light, magnification 
13.) 



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Phil. Trans., A.^ Vol. igs, Plate 4. 



-Etched sheet-lead, x 100. 



Fig. 7. — Freshly crushed lead, x 30. 



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Phil. Trans. ^ A.^ Vol. ig^^ Plate 5. 



Fig. 9. — Same after six days. 



\ 



Fig. 11. — Same after two months. 



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Phil. Trans.^ A.^ Vol igs^ Plate 6r 



Fig 13. — Same after six months. 



\ 



Fig. 15.— Same sheet lead. X 12. 
30 minutes at 200** C. 



After 



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■> c. 



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Phil. Trans. ^ A.^ Vol. igs^ Plate 9. 



Fig. 25.— Same after 40 days at 200^ C. 




is; Fig. 25 (after 40 days at 200^ 



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1 ^s n 



Phil Tram., A.^ Vol igs^ Pl<^te 10. 



nnealiog. X 8. 



Pig. 29.— Tin-plate, etched. X 100. 



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Phil Trans., J., Vol igs, Plate 11, 



I. X 100. 



Fig. 32.— Tin-plate, after 
re-melting the tin and cooling 
quickly, x i. 



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Phil Trans.. A., Vol. igs, Plate 12- 



X 12. 



35. — S>ime after seven dajs at 200° C. 



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Phil Trans,, A,, Vol igs, Plate 13. 



0" a X 8. 



D 



g-. 38. — Weld in Lead, with eutectic. x 30. 



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03 ] 



I in a Magnetic Field. 

D,y F.R.S., and Alfred Hay, B.Sc. 

Read June 21, 1900. 

14-21.] 

jrimental and partly mathematicaF, has 
isional cases of magnetic lines of force 
low of a viscous liquid.* The original 
ms made, showed that the stream lines 
estion, gave results very similar to those 
r the cases of an elliptical and circular 
at the stream lines imder these circum- 
direction of the corresponding magnetic 
ould be used for many practical investi- 
long research dealing with the various 
I extremely laborious, extending without 
a 

ise some method by which a thin sheet 
n could be obtained of any required 
veen two sheets of glass, the required 
e formed. 

w8 connecting the thickness of the thin 
rough it in a given time, so that the 
g to the differences of permeability of 
pertained. 

s undertaken of some cases suitably 
evere a test as possible for ascertaining 
i for any case, accurately, the position 
Ic field. 

ling in succession a very large number 
.n ellipse with the major axis parallel 

Lssociation Report ' (Section A), Bristol Meeting, 

20.12.1900 



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^ LN A MAGNETIC FIELD. 305 

iracteiises the method, is the fact that it 
the lilies of induction not only in air, but 
tei'ial itself. This cannot be accomplished 
1 has therefore been applied to determine 
for a number of cases of mathematical 
plotted by calculation. Inasmuch as the 
f two liquid films with the corresponding 
bove investigation been determined, it is 
e which may be of interest, and some 
)f the method in cases of interest to 

icli has been necessary, has not — as far as 
tble in any published form, and although, 
it, parts of the problem have been dealt 
le account given will be found of use in 



of the relation between the thickness of 
e of flow. 
(Oil with theory. 
)btaining the stream-line diagrams. 

out mathematically and plotted, and its 
>tograph obtained by experiment, 
athematical interest, and also some of 
;trical engineer. 

jstigation of the subject, and 

I elliptic cylinders and con-focal elliptic 



er A, tig. 1, from which it is forced by 
etermined by the pressure gauge B. It 
nt being secured by the lever D which 
E, ^Nvhere its temperature is recorded 
he slide J. 

ss fixed within the frame P ; a well is 
is introduced, a channel being formed 
Eii\ The correct thickness is obtained 
less gauges N, N, N, N, placed at the 



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IX A MAGNETIC FIETJ). 



307 



Film between Parallel Plates. 

20 lbs. pressure. 

wing through in 10 minutes. 
to flow, 
minute. 





B. 


c. 


> 19 


miuutes. 
527 


24 


. 61-6 


162 


51-8 


110 


91 


69-5 


168 


59-5 


94-2 


193-3 


51-8 


97-5 


268-75 


37 3 


125-5 



429 


23-4 


166-5 


557 


18 


210 


861 


11-6 


271 


D78-6 


9-28 


320 


172 


8-04 


329 


!49 


6-05 


416 



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IN A 1 


^tAGNETtC FIELD. 




















f 
















; 


















/ 
















J 






t 










i 


/ 






• 










J 








I 








J 


/ 








\ 








/ 










1 


















^ 






/ 












*> 


/ 


/ 
















^. 


-■ 


















^ 




■— 




B^ 






ILm in /V7C> 
n quduitii 
ftlNE 


7 

has. 

e. 


•O/ 


5 










I 












■ 










A 


1 










1 












/ 




-S 










/ 




1 








/' 




1 






/ 


} 










/ 






1 




i 


/ 










J 










J 


r 












7 
























^ 


--- 


■->. 


B_ 








in . 
I/Ve 


fnch 
n b 
4dLn 


imt 




'i 


><0 



809 



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S^ A MAGNETIC FIELD. 311 

en X — ^ty the velocity at the boundary 

)er unit width ot the layer of thickness 



2 vdx 

Jo 



lin layer between parallel plane walls is 
^er, seen to be proportional to the cube 

experimentally shows a satisfactory 

able us to calculate approximately the 
not, however, well adapted for exact 
it difficulty in measuring accurately the 
f the glass plates are either sufficiently 
eir great thickness) for refined measure- 



ibe of the thickness of the liquid layer, 
xtion of t win give rise to a large error 
1 for the coefficient of viscosity in C.G.S. 
nding to a thickness of '012") the value 
and which may be accounted for either 
Y slight irregularities in the containing 
tre — all of which causes combined might 

:' the viscosity equal to 2*5 ; the density 
in fair agreement with the results of 

out careful absolute measurements, but 
to apply the method to two-dimensional 
! curves obtained (figs. 2 and 3) were 

these experiments whs ordinary tap water, which 
[cation and Lubricants.* 



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IN A MAGKETIC FIELD. 



313 



4. 



Iflmpi 



3 



ew. 



D. 



view. 




O 



} 



^lates and Clips. 



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A MAGNETIC FIELD. 



315 



►vas taken to a place where it could be 
ire given to the sensitive photographic 
Zinc templates of the shapes required 
the wax, the outline was cut with a 
med. In cases where the flow in the 
)ned, and could only be removed after 
-s will be seen in the photographs, figs, 
^ed at all. 



I. 

jtream-line method to the solution of 
decided to work out mathematically 
ction placed in an originally uniform 
3 field, to plot the lines of magnetic 
Moo, and to compare the diagram 
theoretical diagram is given in fig. 9 
agram in fig. 10. In making the com- 

10 was prepared, and fig. 9,"**' which 
s then superposed on it : the coinci- 
)rily established the soundness of the 
\ that slight local divergences along 
stead of the sharp refraction of the 
leoretical diagram, we have in fig. 10 
perfectly straight lines crossing the 
:• or smaller extent, in all the stream- 
It is more marked in those cases 
the two liquid layers, i.e., the perme- 
, is greater. It is clear that the 
l)s the originally uniform distribution 
comparison between the theoretical 
ary to assign to the liquid layer an 
hat of a particular stream-line in the 
lid in fig. 10 is clearly show^n by the 
tlie shape of their boundaries is the 

The method by which the solution 
[ fully explained in the mathematical 



boundaries of the liquid layer are 
means that the diagram gives the 
tion of this paper. 



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IN A MAGNETIC FIELD. 317 

e this latter stream-line diagram with the 
3nt in the shape and general distribution 
triking. 

im for an elliptic cylinder of permeability 
iced in a uniform field with the major axis 
:ing an angle of 45° with the impressed 
for a cylinder of the same permeability, 
. 23 is a similar diagram for a very thin 
How cylinder of fig. 16, but here turned 
ed that fig. 24 confirms the theoretical 
lollow elliptic cylinder bounded by two 
field be uniform. 

I may be treated theoretically as well as 
ses is very limited, and the vast majority 
beyond the powers of analysis. It is in 
ployed by us becomes a powerful weapon 
s of rectangular section, figs. 25 and 26 
* square section placed with one of their 
o it respectively. In fig. 25, where the 
»ng the direction of the impressed field, 
ir are concave outwards ; in fig. 26, on 
a,pidly as we proceed along the field, the 
outwards. This suggests that an inter- 
x)und for which the lines exhibit neither 
Lnd, as a matter of fact, we know of one 
r. Fig. 26 closely corresponds to the 
cently succeeded in working out analyti- 
tiduction.* Figs. 27 and 28 are intended 
bh of one of the sides of the rectangular 
We know from theoretical considerations 
magnetising factor, and thus increasing 
very clearly brought out by a comparison 

3tic fields corresponding to a cylinder of 

inside a hollow square one, and a solid 
teresting in connection with the problem 
s are slightly disfigured by air-bubbles, 
^f, once they are allowed to reach a part 
film varies, 
ry, 1900, p. 225. 



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[ A MAGNETIC FIELD. 319 

^iical Appendix. 

'ptic Cylinders and Confocal Elliptic 

tical theory ot magnetic induction was 
was the first to work out in detail the 
nagnetic material placed in a field of 
id his investigations to the case of an 
it a later date, gave an approximate 
nth its axis along a uniform field. In 
1 ellipsoid of revolution placed in any 
I in solving the same problem for a 
A. G. Greenhill considered the case 

72,\\ diagrams of lines of induction for 
erial placed in a uniform field. On 
;ure in almost every text-book on the 
nown. In Maxwell's great treatise 
iagrams. These include the following 
•netised transversely, and placed with 
)ther (vol. 2, fig. 14); (2) a circular 
5e direction, and placed in a uniform 
;he cylinder is coincident with that of 
al in a uniform field (vol. 2, fig. 15) ; 

uniform field whose direction is at 
1 of the cylinder (vol. 2, fig. 16) ; 
in infinitely long straight cylindrical 
ol. 2, fig. 17). 

iling with the induced magnetisation 
[e obtains a solution by assuming the 
B/r) cos (f>y where r is the distance of 
^r, and (f> the angle between r and 
tants A and B assuming different 

substance, and outside the cylinder 
^7'^ is an integral of the equation 

function Az + B^ of the complex 

t Ibid,, p. 66. 

§ * Journal de Physique ' (1881). 
! Magnetism,' pp. 493-495. 



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IN A MAGNETIC FIELD. 321 

)s of the two media, and itj, Wn for the 
into the coiTesponding media, 
drical shells, it is convenient to abandon 
s, and to have recourse to the circular 



=1 (1) 



> — b^, represents the family of ellipses 

V 

r' > J/ > — a^, represents the family of 



= sin 6, 

~ At 

f, - — =sinhw, 
[b- + I/) 

ellipse, and = constant to a hyper- 

t(> oo ), and (which may vary from 

>osition of a point in the plane of the 

nates. 

I potential function, we make use of 

-^11 that the magnetisation of a solid 

et one of the axes of the ellipsoid be 

lensional case of the ellipsoid to the 

d we see that in this case also the 

might be supposed to be produced 

inary magnetic matter, of volume- 

nt, to be displaced relatively to each 

I of magnetisation, such that pBs is 

the cylinder. If then pY stand for 

cylinders of the imaginary magnetic 

9V 
isplaced cylinders is p -^ Bs. But 



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N A MAGNETIC FIELD. 323 



v^ard-drawn normal at any point of the 



L becomes 

av.- 

putting u = tanh-i - we get for the 



(m-1)'>H (3). 



-.H. 



f. COS — (fi ^ l)ah sinh u cos 0} 

d from these equations. 
le cylinder is obviously ij = constant, 
il lines, we have to determine the 
ce the function which is conjugate 
te to sinh u cos is cosh u sin 6, 



— T — sinh?4 
l)ao 

, K being a constant which varies 

nipressed field is along the minor 
>ii to the lines of the external field 

- , Sinn ?/, 

le line to another. 
elliptic cylinder may be extended 



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IN A MAGNETIC FIELD. 325 

n w = u^, Vi = V^ ; this gives 
t + fcB, . (4) 

A^ cosh t^i — Atf sinh Uj (5). 

Si( ^ Bit ' 

. — jxbA^ — /jtaBj . . .... (6), 

I — fiA^ sinh u^ — /aB cosh i^j . . . (7) 

A, Ai, B^, and A^, we get 

^cosh^i) , — im 

- l)a6(/Lico8hi^ + sinhi^^) * ^^ 

sh 2c^ 4- sinh Ui) , i \ TT 

l)aJ(yt6C0sh?<i + sinhwi) ' ^^ 



a somewhat simpler form by putting 
t^y 6i stand for the semi-axes of the 
ering that u^ = tanh *" ^ (&i/«i), we get 

TT • H 



^a7}(-^) 



— Oft) 
- «i6) 









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N A MAGNETIC FIELD, 327 

space, 

the substance of the shell, and 

external space. 

the view of plotting curves, it is con- 
! cosines, such as the one compileil by 
?al Society of London. Corresponding 

I easy step to pass to Cartesian rect- 
it for plotting the r^urves. 

mentioned, be extended to any number 
il, however, wlien numerical data are 
>uce in the equations instead of first 

lis may be regarded as a limiting case 
^coming equal. If in the expressions 
e case of a hollow elliptic shell we 

II proceed to the limit h = a, we find 
i\'ith those deduced by Du Bois for a 
netic Shielding."* But although an 
cial case is thus obtained, it is much 
lells, to follow the method developed 



S Seddon 2, p. 613. 



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PhiL Trans,, A,, Vol, 195, Plate 14. 



10 

im-line Diagram corresponding to 
iheoretical Diagram of Fig. 9. 



12 

inite Elliptic Cylinder in uniform field, 
^atio of axes 3:1. Permeability 1000. 



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Trans,, A., Vol. ig^, Plate 15. 



14 

Cylinder in uniform field, 
ability =100. 



16 

>tic Cylinder in uniform 
ieability = 100. 



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il. Trans., A., Vol. \g^, Plate 16. 



18 

iric Shield enclosing Solid 
. Permeability =100. 



20 

ar Cylindric Shield. 
line Diagram, 
-ability = 100. 



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Phil Trans., A., VoL 195, Plate 17. 



22 

r. Ratio of axes 3:1, major 
it 45^ to field. Stream-line 
. Permeability =100. 



'24: 

iptic Cylinder. 
lined 45° to field. 
>ility=^100. 



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Trans., A,, Vol. 195, Plate 18. 




of Square Section. 
lity=100. 




Rectangular Section. 
lity=100. 



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Trans., A., VoL 195, Plate 19. 



30 

of Triangular Section, 
ability =100. 



3:i 

blinder enclosing Solid 
Permeability = 100. 



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'ns.,A., Vol, \g^y Plate 20. 



) and Teeth of Toothed - 
Permeability =100. 



6 

Lture by Pole-piece. 



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PhiL Trans., A., Vol. 195, Plate 21. 



31) 

of Magnetic Field near edge of 
ole -piece in Dynamo with 
Toothed -core Armature. 



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\egrals to Optical Problems. 

\ Isaac Netvton Student in the 
Ige. 

HOMSON, F,RS. 



, 1899. 



Page 

notions 331 

>red 331 

..,.•.... 331 

......... 332 

333 

334 

335 

ar a length of time which 
)ing detected .... 335 
»e estimated by summing 
to which it is resolved 
335 

theory of widening of 
vibrations, Thomson's 

338 

by Lord Eayleigh. . 339 

3f the pulses .... 339 
o light of composition 
i pulse, subject to the 
the sDoiallest observable 

waves 339 

itical expression gives 

342 

342 

cZ hy a law 343 

24.12.1900. 



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J. PROBLEMS. 331 

Page 
360 

>chromatic light will arouse 
reasing intensity .... 361 
[Kr's integral 361 



some progress with the mathe- 
mpose natural light, 
e phenomena forbid us to regard 
iin of simple waves, such as may 
ne, the equations of optics find 
iesirable to enquire how far we 
)f simple wave-trains by means 
procedure was first suggested by 

permissibility of this process, 
and independence of the simple 
be found an attempt at a strict 
ciples (i.) that we are cognisant 
luced by the light during an 
the detector in use (the eye, a 
ned with simple wave-lengths, 
integrated energy we observe, 
as been put forward with great 

md plane-polarised light. 

:ter than by a quotation £rom 

srplications qu'elle donne des 
vement simple, dans lequel la 
• una Equation de la forme 



cements of an elastic medium, or of 
sctors, which can be interpreted in 



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:jal problems. 333 

efining it for a finite interval alone. 
les of the time, from — oo to + oo. 



in nt)du 
fiy) sin uvch\ 

a 

f elementary simple vibrations, of 

Ht) 

results of the undulatory theory 

all values from zero to infinity, 
nber of simple circular functions 
have meaning, as in the familiar 
nstruments, &c. The question we 
g in the limit, when their number 

1 ; this was, in fact, offered by 
ach of the component vibrations 
all time. This is true whatever 
ng. But this disturbance may, 
? interval of time. Take the case 
I. PoiNCAR^, will separate the 
ed separately. Hence a spectro- 
time before it is kindled, and for 
Is must therefore be fallacious. 

spectroscope possesses infinite 
telescope will be illuminated by 
le matter from another point of 

rruji'ing in Physical Problems. — In pure 
Is subject to certain limitations. These 
luities, or infinite sets of fluctuations, or 
d neither infinities nor discontinuities. 
commonly used to represent physical 
t character of the method ; a function 
cjucstion would be without infinite or 

95 ; also see Schuster, *C. R./ 120, 



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AL PEOBLEMS. 



335 



K - ^sM^- 



3 upon the distribution of energy 
■ phases of corresponding elements 

order to prove that, in case of a 

e amount of interference depends 

any assumption respecting the 

onsequences. 

•nfined to the particular case of 
perceptible fluctuations or other 

I the long waves of Hertz is by 
jreat compared with the periods 
sed to perceive and register the 
r chemical effect, photographic or 
minescence which the radiation 
le features of a single wave can 

ht, with a view to discriminating 
lely with the integral effect over 

isider the molecule as a simple 
istified, we may prove that the 
ling but the partition of energy 
:er integral. The phases will of 
it 
id. 



or no ; but, if that condition 



+• 



i/(«) depends upon ^f{t)dt 

— 00 

qual to 



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AL PROBLEMS. 337 

'erent periods drop out (a familiar 
ite of absolution is dependent on 



►rptiou. So again with physiolo- 

s produced by light we probably 
le of luminescence also. It may 
* dissociation. But, if the disso- 
excited vibrations become large, 
ihtle88 some molecules will split 
iile the precise timing of its own 
be all-important. But on the 
f dissociation will perhaps depend 
ifying the average structure, 
linear equation for the vibration 
3t step towards a solution of a 
>KES, ''Linearity applies to the 
} ether — but it does not follow 
omplex system of molecules." 
t treatment to the spectroscopic 

en direction is compounded of 



e of the instrument, and the 
^ tff — depends upon the same 
ip — depends upon i/r.. 



rznining the constancy of the 
t is constant, so also is the 
iiy ; hence, the spectroscopic 



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PROBLEMS. 339 

urve to completely specify the 
ts will be justified below. 
.EiGii* in his paper on "The 
proposes to regard this as an 



(8) 



(22) 



% is therefore C^e'^'^^du. 

necessarily equal to (8), and of 
le range from — oo to + cx), the 

value of u would be indeter- 
could be assigned ; and if the 
iltant would again be indeter- 
; there is room for an infinite 
ioncerned only with an average, 
to be. proportional to the total 
phase-relations. In the aggre- 
^y is distributed is still, for all 

theory. But when we come to 
are certain questions which still 

is confined to a certain small 
ssor Thomson's theory, how 
;ive the properties which Lord 
Again, we might suppose that 
egrees of crowding among the 
b a great number of them are 
)e so thinly scattered as to be, 

occupied by each. Experiment 
1 correct ; we may enquire what 
rd Rayleigh's theorem to the 

us, be equivalent to a spectrum, 



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PROBLEMS. 341 

base of the resultant will be 



c motion A. 

lall, the new phases will differ 
leat, the new phases will differ 
11 have no apparent connection 

radiation. First, we can only 
)h we have taken to be T ; we 
pulses in T. Now Schuster's 
dvgj of a radiation, and its ex- 
i-Ier expression for the resultant 



ion of V, but partakes of the 
this point we make use of 
1 particular wave-lengths, but 
ave-length. Bearing in mind 
>es, we have a spectrum whose 



Lirve will be less rapid as we 
)ave just seen, when the time 
ms are so crowded as to be 
the meancurve n(f}%u). But if 
e of angle included in the set 

se intermediate between the 
e pass continuously to quicker 
) and so on ; the divergences 
t on a broad and theoretically 

miount of energy in the slow 
)f Sound,' ed. 1894, p. 40. 



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niOBLEMS. 343 

L much lai'ger exposure, or as 

are not all similar is obvious. 
es of constant displacement^ 
inie time the proportions of 
tween x and x + dx are, say, 
ulses ; suppose that they give 
* the mixture is 



jight. 

)n by supposing it to consist 
is practically a pulse of con- 
thickness of a pulse is com- 
r the cathode stream. Lord 
»e regarded as simple waves 
)v the properties of a succes- 
mcy of statement. Professor 
new. He has held that the 
Now this is a valid objection 
rht of any composition what- 
sess definite phase-relation ; 



)f the present essay that the 
truna of definite composition, 
of being brief in comparison 
1 be a great number of them 
iverages over small ranges of 
es of time- and wave-length 
cession of pulses and that 
3he single pulse. 
1 the scale of Avave-length. 
of X which is zero from 



CO 



y, 1898. 



•, 1898. 



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>ROBLEMS. 345 

ignetic force great and nega- 
legative are to be so balanced 
pulse shall vanish. On this 
IS, Sir George Stokes* bases 

be zero amplitude for the 
lergy in the visible spectrum 
wfU be of order E^d . {d/X^y, 
stead of 10"^. Now diffrac- 
j order ; very much shorter 

and in any case would give 
posed forms will be sensibly 
3h higher degree than those 

IS, 

nber of independent pulses 
an incandescent gas. We 
I, and to have no visible 
tion as composed of plane- 
Furthermore, the amount 
ected by absorption. The 
free vibration, 
omogeneous. One reason 
les in the line of sight will 
doubtless be the altered 
This will perhaps become 
r it at present, 
iie train of single waves 
gth, but has a definite 
ed below. The Doppler 
arrive at the remarkable 
3 pressure is indefinitely 
ted by Lord Rayleigh, 

ion virith another. The 
3ction. The vibration 
>re be suddenly and 
otal radiation received 

)7. 



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PROBLEMS. 347 

ig off from a maximum at k 

(i) 

), where 7i is the distance from 
ave-length. 

cs both Athwart and in Line 

imum brightness, and definite 

erent lengths of train. 

v^ith velocity v, a fraction C-^'* 



'Jr"4 



./p1 



("•). 



Jume, 

3ity Vy a fraction 



of waves of length between 
crht. 



Lction ^e ^ dr^ will give free 



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L PROBLEMS. 349 



hf^'^mi- 









igh the breadths of elementary 
•tly, yet the application of his 
yte detailed information. The 
retardation u of the two half 
eir " visibility " estimated for 
isibility-curve thus constructed 
m line, and find out something 

intensity of light for position x 



i-difference. 
nd 34. 



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PROBLEMS. 



351 



ble. We axe enabled, however, 
ves in his paper on the kinetic 



,-p« 



< 



4- 1) e-^dP 



notation. 

e values in column 6 of Tait's 
^ the function is very near to 



•- 


1 1 


1 1 i 1 1 


1 — 




1 


I 1 1 1 i 






















r 








































































^. 


















\^ 




















^^ 


s. 


















^.:^ 


^;>v 




















'"'••., 


=^ 


^ 







£-0 



£•3 



>se /x so that the two curves 



; graphically we see that we 



■^z/^2 > II. (dotted curve) is 
he integral through using 11. 
s of the order of the errors 
t^pare our results. 



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)BLEMS. 353 



esent paper necessitates a 
tted from Lord Eayleigh's 
above-mentioned necessity 
theory. The conclusion of 
its for a certain fraction of 
rom f to ^ for the different 

iby 



I 



the spectrum on a scale of 
11 be Sq. Hence the " half- 
onnected by the relation 



to the visibility half-width 
should be diminished by a 

md collisions comparatively 
LS would disappear, the free 
:pected that the formula for 
rd Rayleigh, in which the 
g is valid, there is no such 
indefinitely reduced; the 
It is noteworthy that the 
oes Lord Rayleigh. This 
tation. 

lathematically homogeneous 
certain continuous spectrum, 
yrsis that, for zero pressure, 
r curve for the total spectral 



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)BLEMS. 355 

iomponent curves are, the 
lUow ourselves to substitute 
these become narrow, 
ied for r, will not affect the 
th of train is taken to be 
ater velocities give shorter 
)ther ; for it gives too great 
dties, which velocities send 
of the error is to make the 

he limiting width for zero 
, and then to proceed to the 

me. 



rum Lines. 

3 vibrations of an atom are 
umple gas will be to some 
loped by JAUMANN.t 
ibration 

(x.) 

IS been pointed out in the 
) physical meaning for the 

lent emissions of this form 

* a gas cannot strictly be 
(x.) will never be allowed 
a stage by a new collision. 
)f a naolecule has generally 
shall not be making a 
ives. We will proceed to 
^s so rapid as to allow this 

application of Fourier's 



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OBLEMS. 357 

mogeneous Light 

oducing a large number of 
f a certain regularity in the 
las been abundantly refuted 
)t be produced without the 
iges is an index, not of the 
)f the spectroscope, 
out a spectroscope, by using 
nogeneity. The number of 
case, it is also a test of the 

y a Fourier integral of the 



ill fluctuate rapidly in terms 
uency p^ we will use the 

values of u small compared 

ut + \lf)du. 

to a region on either side of 



- \lj)du 

with those of oospt. The 
Biiote a simple vibration of 
of the expression will be, 
[)ply to 



e Theory/' « Encycl. Brit.'; ' Phil. 
^94 



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MS. 



359 



ot is of the same order : 
=i. In accordance with 
3^ wjwe-lenjjths ; while 
Hjueiicy of the light 



a naturtil nxdiatioii {see 

:er of this ti'eatmeiit of 
has a historiciU inteivst, 
may he worth while to 



vibrator is 8oon to dopond 

The effect of tlie irrogii- 
^ning of the Rpectruni line, 
svhich laid the foundatioiiH 

TU) natural radiation ih a 
t of the irn^gularitit^H upon 
it of tlie natural vibratiouH 
ukritieH in the light would 

K 520. 



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[S. 361 

iiiiig of the incident 
11 of a large number 

of the amplitudes, 
nil of the component 
iiergy of the natural 
•ns ; in other words, 

will become greater 
there is no tendency 
interact one another. 
jes are, so to speak, 

IS light will give a 
e^being entirely due 

1 be prevented from 
ponent trains would 
ation settles down to 
5 irregularities, 
plication of Fourier. 



:egrand involves an 
re that the integral 



I- We are forced to 
s to the actual pro- 
he natural vibration 
the observed effects 

•n JR^rfu will excite 



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t/ Marble. 

rqfessor of Gex>logy in 
M.Inst.C.E., Head of 
lical Si'fiool {foi^ierly 

y). 

R.S. 



Page 

363 

369 

370 

370 

373 

376 

jr . . . . , 382 
tificial Deforma- 

386 

tificial Deforma- 
lighly contorted 

387 

398 



I in many parts of the 

manner is a fact which 

a glance at any of the 

ich have been prepared 

the hardest rocks have 

' or '* flow " of material 

e facts are undisputed, 

lug, has taken place is a 

5.1.1001 



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V OF MARBLE. 367 

calcite, however, wais 
which existed in the 
ind cracks, as well as 
a finely pulverulent 
portions of the mass, 
jsure. GuMBEL con- 
;ticity on the part of 
[)f the alahaster, and 
In the field, that the 
re they had become 
hows distinctly under 
to a recementation of 
*. It was shown by 
e in the case of the 
d to powder; in the 
^ed that deformation 
for the phenomenon 
jliaracters and optical 
I been submitted to 



)ut by PFAFF.t He 
lofen in a steel block, 
V n upon it. A very 
estone, and this was 
e amounting to 9970 
-as not displaced, and 
specimen of the same 
ure of 21,800 atmo- 
le limestone did not 
left on the polished 
the conclusion that 

y different materials, 
5uts reproduces very 
; described ; but the 
red out to receive a 
• the hole a steel die, 



ire, 



■ Zeit. des Vereines 



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OF MAEBLE. 369 

Jans of explosives. 
Tmmeter of about 
od together again 
chamber in which 
pJetely perforated 
' the influence of 
what plastic, the 
e when removed, 
e marble had lost 
says, it was seen 
into a solid mass. 
)und the central 

of rocks (and it 
iJ work in which 
reared up to the 
3 and in certain 
taken in any of 
applied, of the 
w cases, of the 
5cription of the 
strength of the 
:ie pressure can 



ing about the 
s crust, where 



ssion will not 
ce molecular 
niovement to 
assure affects 

k ^virkt, der 



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OF MAEBLE. 371 

mit combined with 
1 however it was 
^ tubes of wrought 
nanoe by rolling a 
le strip to the bar 
is then bored out, 
uarter of an inch 
the tube instead 
he requirements 
) of the marble, 
in length, were 
ff, of Gottingen. 
ajs accomplished 
he tube, and so 
tube when cold. 
pass completely 
)n allowing the 
aed, and it waa 
3red indispens- 
; applied, as it 
le experiments 
>r immediately 
an accurately 
3 was applied, 
ising a double 
accompanying 

ist of square 
)osite to it by 
5 marble with 
being kept in 
to cylindrical , 
them when 
in diameter, 
las its upper 
3tion to the 
rrosion and 
[na are cen- 
sures to the 
pper end of 
lure is kept 



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W OF MARBLE 873 

ly bulge. This bulge 
rounding the marble. 
igns of rupture, when 



rratures. 

at ordinary tempera- 
Perent cases, the con- 
ns taking place more 
ninutes to 64 days, 
ceased, and in this 
The final amount of 
ved signs of rupture 

a close. 

imn before the pres- 
) completion of the 
y slowly i the time 

ble, consistent with 
;, and the tube was 
ling machine along 
as found to be still 
5, now completely 
. without mechani- 
in between them, 
itting the marble 
,. tube adhering to 

1 the former case 
vhile in the latter 

or two instances 
le latter a smart 
^ that it could be 
e exterior surface 
;h and conformed 

all the fine tool 

y from the tube 
r the dimensions 

with equal case 



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V OF MARBLE. 873 

Y bulge. This bulge 
'ounding the marble, 
gns of rupture, when 



ratures. 

it ordinary tempera- 
erent cases, the con- 
s taking place more 
linutes to 64 days, 
ceased, and in this 
The final amount of 
red signs of rupture 
3 a close. 

mn before the pres- 

completion of the 

y slowly, the time 

Je, consistent with 
, and the tube was 
ling machine along 
3ts found to be still 
I, now completely 

without mechani- 

in between them, 
Itting the niarble 

tube adhering to 
1 the former case 
^hile in the latter 

or two instances 
^e latter a smart 
y that it could be 
e exterior surface 
th and conformed 
. aU the fine tool 

y from the tube 
f the dimensions 

, ^ith equal ease 



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y OF MARBLE. 375 

extent by the cones 
as above mentioned, 

h the unaltered cone 
en examined under 
h had taken plia.ce. 
r its turbid appear- 
aic of the unaltered 
in Experiment P of 
ved and branching 

through the rock. 
)andsof very small 
place. The calcite 

so produced have 
fter the movement 
mbers of irregular 
have been carried 

The structure is 
Ispars and many 
ble showing this 
ty and magnified 

'067 inch, which 
it bulging. The 
nulated portions 
itinctly twinned, 

movements of 
be seen to have 
attening of the 
have been bent 
it cases, whichi 

polysynthetio 
ihe destruction 
I of the calcite 

essure to alter 
lently due to 
111 be referred 

of somewhaj; 
-this structure 



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OF MARBLE, ^77 

ice (E). The whole 
Bunsen flame. The 
learlv {\s possible at 
le extreme limits i>f 



temperature 

!y hiH jKwgeiblo 
inn ill inches 



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OF MARBLE. 381 

ut unfortunately it 
which rendered it 
n contact with the 
around the central 
'here had been no 
ered marble tested 
When sliced and 
caclastic structure, 
3k a foliation which 
ed the very narrow 
scribed. The twin 
by strain shadows, 
nor very striking, 
into very irregular 
irms (fig. 6), quite 
The individual 
lounced movement 
* twin lamellae, is 



3 



the calcite indi- 
ke outline of the 
twin lamellfe. In 
I inward between 



1 interest, as it is 
over one another 



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F MARBLE. 383 

•med while at a 

ier a pressure of 

slowly and at as 



2 mouths. The 



Ui-ushing load after 
deformation. 



longer than the ori- 
ginal rock 



o 25*51 per cent, 
in former experi- 



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W MARBLE. 385 
leformation is not 

1 and some trans- 
ous mosaic before 
li section, starting 
>ward the middle, 
ones. Under the 
amber of fine and 
ilong which there 
vse, and elsewhere 
aclastic structure. 
)n may be said to 
e, accompanied by 

seen in the case 
0^ C. The calcite 
al (none are more 
rmed rock a very 

flattening of the 

long as they are 
but no twinning, 
aning, givbg rise 

will show strain 
c twinning at the 
a when magnified 
lamellae in several 
of between '0005 

I rock, it appears 
the calcite grains, 
above described, 
even in this iron- 

the deformation. 
1 of infinitesimal 
hus contributing 
on, however, are 



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OF MARBLE. 387 

1 three planes or 
Q to accommodate 
ids. The action is 
sUps, the portion 
olid. The process 
igate effect is not 

Lination in metals 
he crystal on one 
) need to suppose 
dip when it occurs 
because the metal 
, and is able to be 
idividual crystals, 
its shape and its 
occurring within 

h those presented 
at the agi'eement 
7 applicable to the 
Inscribed, as it is of 
:al is squeezed flat 



ARBLE BY ArTI- 
LlMESTONES AND 

J Crust. 

h s crust has been 
lalf-century, com- 
of limestones and 
Hy with unaltered 
ing from pressure, 
mites from many 
sections of these 
Neozoic age, does 
served, and states 
seen, producing a 
[>s, one of them a 

er K. Bayer. Akad. 



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OF MARBLE, l\H\) 



a. 

la. 

^anada. 

Janada. 
, Canada. 



)tful Origin. 



SCructureif. 



la. 

la. 

I. 

uada. 

la. 

/anada. 



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f 

k 

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a 
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ELS 

of 

be 

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he 

st- 

tits 

}tic 

% 

lOSt 
'OW 

her 
i of 
;up 

axe 
^hat 
ight 
how 
In 
from 
ding 
med 

and 
rmed 

age 
ouble 
( the 
labby 
?en to 
)arely 
jalcite 



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t 

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e 
a 
a 
i 
i 
i 
s 
t« 
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